02-DifferentialCalculus.Rmd

# (PART)  Differential Calculus {-}


# Exponential and Logarithmic Functions 



## Lecture Content

[**Video:  Logarithms**](https://youtu.be/PCHQAFFvGLc)

[**Video:  Quadratic Functions and Models**](https://youtu.be/frWepga5lgc)

[**Video:  Exponential Functions and Models**](https://youtu.be/qV-HYR-iBSs)

[**Video:  Logarithmic Functions and Models**](https://youtu.be/nBoc5u7ucFU)



---


## Lecture Notes

### Rules for Exponents

Let \( a > 0 \), and \( m, n \in \mathbb{R} \):

- \( a^m \cdot a^n = a^{m+n} \)
- \( \frac{a^m}{a^n} = a^{m-n} \)
- \( (a^m)^n = a^{mn} \)
- \( (ab)^n = a^n b^n \)
- \( a^0 = 1 \)
- \( a^{-n} = \frac{1}{a^n} \)


---


### Rules for Logarithms

Let \( a > 0 \), \( a \neq 1 \), and \( x, y > 0 \):

- \( \log_a(xy) = \log_a(x) + \log_a(y) \)
- \( \log_a\left(\frac{x}{y}\right) = \log_a(x) - \log_a(y) \)
- \( \log_a(x^n) = n \cdot \log_a(x) \)
- \( \log_a(a) = 1 \)
- \( \log_a(1) = 0 \)

---

### Finding an Exponential Equation from Two Points

To find \( f(x) = C \cdot a^x \) that passes through points \( (x_1, y_1) \) and \( (x_2, y_2) \):

1. Set up the system:
   \[
   y_1 = C \cdot a^{x_1} \\
   y_2 = C \cdot a^{x_2}
   \]

2. Divide the two equations to eliminate \( C \):
   \[
   \frac{y_2}{y_1} = \frac{a^{x_2}}{a^{x_1}} = a^{x_2 - x_1}
   \]

3. Solve for \( a \):
   \[
   a = \left( \frac{y_2}{y_1} \right)^{\frac{1}{x_2 - x_1}}
   \]

4. Plug back into one equation to solve for \( C \).

---

### Solving Exponential and Logarithmic Equations

#### Exponential Equations{-}

To solve \( a^{f(x)} = b \):

1. Take logarithms of both sides:
   \[
   \log(a^{f(x)}) = \log(b)
   \]

2. Use the power rule:
   \[
   f(x) \cdot \log(a) = \log(b)
   \]

3. Solve for \( x \).

---

#### Logarithmic Equations {-}

To solve \( \log_a(f(x)) = b \):

1. Rewrite in exponential form:
   \[
   f(x) = a^b
   \]

2. Solve the resulting algebraic equation for \( x \).

---

### Applications and Formulas

#### Continuous Compounding {-}

If interest is compounded continuously:

\[
A = Pe^{rt}
\]

Where:
- \( A \) = final amount
- \( P \) = principal
- \( r \) = annual interest rate
- \( t \) = time in years
- \( e \approx 2.718 \)

---

#### Total Revenue from Demand Function {-}

If price \( p(x) \) depends on demand \( x \) (e.g. \( p(x) = 120 - 3x \)), then:

\[
R(x) = x \cdot p(x)
\]

Where:
- \( R(x) \) = total revenue
- \( x \) = quantity sold
- \( p(x) \) = price per unit as a function of demand

You can expand and simplify to analyze maximum revenue using calculus or vertex form (if quadratic).

---


##  Examples


### Quadratic Functions

[**Video:  Solutions **](https://www.youtube.com/watch?v=dsS6JHuwyzA)


1. Identify \(a\), \(b\), and \(c\) in the equation below, then solve using the Quadratic Formula.
\[
x^2 + 18x + 80 = 0
\]

2. Determine the number and type of solutions for the equation:
\[
-2x^2 -9x -9 = 0
\]

3. The equation
\[
5x^2 +15x +4 = 0
\]
has two solutions.  Find them.

4. Consider the quadratic function:
\[
f(x) = 9x^2 - 1
\]  Find the:
   - Vertex
   - Largest \(x\)-intercept
   - \(y\)-intercept

5. Consider the parabola given by:
\[
f(x) = -4x^2 -8x +14
\]
Answer the following:
   a. The graph of this function opens: ☐ Up ☐ Down  
   b. Domain (interval notation):  
   c. Range (interval notation):
   

6. The Acme Widget Company has found that if widgets are priced at \$218, then \(14,000\) will be sold. For every increase of \$13, there will be 300 fewer widgets sold.  The marginal cost is \$130.80 per widget. Fixed costs are \$12,000. If \(x\) represents the price of a widget, find:

   a. Number of widgets sold
   b. Revenue
   c. Cost of production
   d. Profit
   e. Price that maximizes profit
   
   

---



### Exponent Rules

[**Video:  Solutions **](https://www.youtube.com/watch?v=V3n2uVAkdn4)


1. Simplify the expression completely:  

\[
\frac{x^{22} \cdot y^{79}}{x^{16} \cdot y^{20}}
\]

2. Simplify the given expression:  

\[
\frac{(a^4 z^2)^4 \cdot (-a^2 z^2)^6}{\left( -a z^3 a^5\right)^2}
\]



3.  Sketch each of the following:

   a. \(4(0.65)^x\)  
   b. \(3(1.17)^x\)  
   c. \(4(1.17)^x\)  
   d. \(4(1.52)^x\)  
   e. \(4(0.88)^x\)  
   

4. Use the like-bases property and exponents to solve:  

\[
\left( \frac{1}{5} \right)^{x+7} = 5^{7x+4}
\]


5. The population of the world in 1987 was 5 billion and the annual growth rate was estimated at 2% per year. Assuming an exponential growth model, find the projected world population in 1992.




---




### Logarithm Rules

[**Video:  Solutions **](https://www.youtube.com/watch?v=c5rwovJLt24)


1. Express the equation in exponential form.

  a.  
\[
\log_{2} 8 = 3
\]  
That is, write your answer in the form \( 2^A = B \).  

   b.  
\[
\log_{5} 3125 = 5
\]  
That is, write your answer in the form \( 5^C = D \).  


2. Simplify:  
\[
\log_{z} \left( \left( \frac{1}{z} \right)^{6} \right)
\]

3. If \(\ln(a) = 2\), \(\ln(b) = 3\), and \(\ln(c) = 5\), evaluate the following:

   a. 
\[
\ln\left( a^{a} b^{-2} c^{-2} \right)
\]

   b.  
\[
\ln\left( \sqrt{b^1 c^{-2}a^{-4}} \right)
\]

   c. 
\[
\frac{\ln\left( a^{-4} b^{4} \right)}{\ln\left( \frac{b}{c} \right)^{2}}
\]

   d.
\[
\left( \ln c^{-2}\right) \left( \ln \frac{a}{b^3}\right)^{4}
\]


4. Find the domain of:
\[
y = \log(5 - 6x)
\]  

5. If \( e^{7x} = 27 \), then $x$ is what?


6. Solve for \( m \) in the equation below. 
\[
2 \log_{8}(m) + 4 = 8
\]  


7. Solve for \( x \):

\[
\log(x+5) - \log(x+3) = 1
\]  


8. A culture of bacteria grows according to the continuous growth model:
\[
B = f(t) = 500 e^{0.073t}
\]  
where \(B\) is the number of bacteria and \(t\) is in hours.

   a. Find \(f(0)\)  
   b. To the nearest whole number, find the number of bacteria after 5 hours.
   c. To the nearest tenth of an hour, determine how long it will take for the population to grow to 1100 bacteria.



---




## Practice Problems



1. Find the equation of the exponential function that passes through $(2,-4)$ and $(4,-16)$.

2. Solve $4^x = 3$.

3. Solve $4(1.5^{2x-1}) = 8$.

4. Solve $e^{2x} - 9e^x + 20 = 0$.

5. Why is the logarithm of a negative number not defined?

6. Sketch $f(x) = \log_5 x$ and $g(x) = 5^x$.

7. Suppose $\log(a) = 5$, $\log(b) = 3$ and $\log(c) = -2$. What is $\log\left(\dfrac{ab^4}{c^7} \right)$?

8. (Applied) How long will it take a \$500 investment to be worth \$700 if it is continuously compounded at 15\% per year? (Give the answer to two decimal places.)

9. (Applied) The Better Baby Buggy Co. has just come out with a new model, the Turbo. The market research department predicts that the demand equation for Turbos is given by
\[
q = -2p + 320,
\]

where $q$ is the number of buggies it can sell in a month if the price is \$p per buggy. At what price should it sell the buggies to get the largest revenue? What is the largest monthly revenue?



---


##  Self Assessment

Time yourself and try to solve the following questions within twenty minutes.

1. Find the equation of the exponential function that passes through $(3,5)$ and $(4,25)$.

2. Solve $6^{3x+1} = 30$.

3. Suppose $\log(a) = 2$, $\log(b) = -7$ and $\log(c) = 3$. What is $\log\left(\dfrac{a^2b^3}{c^5} \right)$?

4. The median price of a home in the United States declined continuously over the period 2005–2008 at a rate of 5.5\% per year from around \$230 thousand in 2005. Write down a formula that predicts the median price of a home $t$ years after 2005. Use your model to estimate the median home price in 2007 and 2010.

5. In 2005, the Las Vegas monorail charged \$3 per ride and had an average ridership of about 28,000 per day. In December 2005 the Las Vegas Monorail Company raised the fare to \$5 per ride, and average ridership in 2006 plunged to around 19,000 per day.

   a. Use the given information to find a linear demand equation.  
   b. Find the price the company should have charged to maximize revenue from ridership. What is the corresponding daily revenue?  
   c. The Las Vegas Monorail Company would have needed \$44.9 million in revenues from ridership to break even in 2006. Would it have been possible to break even in 2006 by charging a suitable price?

---




## Lesson Checklist 


This checklist is designed to help you keep track of what you need to work on. The main goal is to be aware of what you need to focus more attention on. Place an $X$ in the appropriate box beside the skill below. 
\bigskip

\noindent
\begin{align*}
&\textbf{Developing (D):} &&\textrm{You still need to work on this skill.}\\
&\textbf{Consistent (CON):} &&\textrm{You use the skill correctly most of the time.}\\
&\textbf{Competent (COM):} &&\textrm{You show mastery of the skill.} 
\end{align*}


| Skill                                                         | D  | CON | COM |
|---------------------------------------------------------------|----|-----|-----|
| Simplify expressions involving exponents or logarithms.       |    |     |     |
| Find the equation of an exponential function through two points.|    |     |     |
| Solve exponential or logarithmic functions.                   |    |     |     |
| Maximize the total revenue function, given a linear demand curve.|    |     |     |
| Solve problems involving the continuous compounding formula.  |    |     |     |
| Solve applied problems involving exponentials or logarithms.  |    |     |     |











# Limits and Continuity




## Lecture Content

[**Video:  Numerical and Graphical Approaches to Limits**](https://youtu.be/FPlR88H9_A0)

[**Video:  Limits and Continuity**](https://youtu.be/eloHCkPZwos)


---



## Lecture Notes

### Estimate a Limit Using a Table of Values

To estimate a limit using a table:

- Choose values of \( x \) that get closer to the target from both sides (left and right).
- Observe how the function values \( f(x) \) behave.
- If the values approach a single number, that is the estimated limit.

**Example:**
If \( f(x) = \frac{x^2 - 1}{x - 1} \), use values close to 1 (e.g., 0.9, 0.99, 1.01, 1.1) to estimate \( \lim_{x \to 1} f(x) \).


---


### Use Factoring to Evaluate Limits

Factoring helps eliminate indeterminate forms like \( \frac{0}{0} \).

**Steps:**

1. Factor the numerator and denominator.
2. Cancel common factors.
3. Substitute the limit value.

**Example:**
Evaluate \( \lim_{x \to 2} \frac{x^2 - 4}{x - 2} \):
\[
\frac{(x - 2)(x + 2)}{x - 2} = x + 2 \Rightarrow \lim_{x \to 2} = 4
\]


---


### One-Sided Limits and Vertical Asymptotes

- **Left-hand limit:** \( \lim_{x \to a^-} f(x) \)
- **Right-hand limit:** \( \lim_{x \to a^+} f(x) \)

If the function approaches infinity or negative infinity near a point, it may indicate a vertical asymptote.

**Example:**
\[
\lim_{x \to 0^-} \frac{1}{x} = -\infty,\quad
\lim_{x \to 0^+} \frac{1}{x} = \infty
\]


---


### Dominant Term Analysis for Limits at Infinity

To evaluate limits as \( x \to \infty \), focus on the **dominant (highest degree)** terms.

**Steps:**

1. Identify the highest power term in the numerator and denominator.
2. Divide all terms by the dominant power of \( x \).
3. Simplify and take the limit.

**Example:**
\[
\lim_{x \to \infty} \frac{3x^2 + 5}{x^2 - 2x} = \lim_{x \to \infty} \frac{3 + 5/x^2}{1 - 2/x} = \frac{3}{1} = 3
\]


---


### Continuity of Piecewise Functions

To determine if a piecewise function is continuous at a point:

1. **Check function is defined:** \( f(a) \) exists.
2. **Evaluate both one-sided limits:** \( \lim_{x \to a^-} f(x) \) and \( \lim_{x \to a^+} f(x) \).
3. **Check limit equals value:** \( \lim_{x \to a} f(x) = f(a) \)

If all three conditions are met, the function is continuous at \( x = a \).

**Example:**

\[
f(x) =
\begin{cases}
x^2, & x < 2 \\
4x - 4, & x \geq 2
\end{cases}
\]

Check continuity at \( x = 2 \):
- \( f(2) = 4 \)
- \( \lim_{x \to 2^-} = 4 \), \( \lim_{x \to 2^+} = 4 \)
- Since limits and value agree: **continuous**



---


## Examples


### Limits

[**Video:  Solutions **](https://youtu.be/QiA2DBzh0YU?si=agk2U7DCaVmjMCPm&t=143)


1. Find the following limit:

$$
\lim_{x \to 7} \frac{x^3 -3x^2 - 29x +7}{x - 7}
$$


2. Estimate the limit numerically or state that the limit does not exist:

$$
\lim_{x \to 121} \frac{\sqrt{x} - 11}{x - 121}
$$

3. Use numerical or graphical evidence to determine the left and right-hand limits of the function.

$$
\lim_{x \to 4^-} \frac{|5x - 20|}{x-4}
$$

$$
\lim_{x \to 4^+} \frac{|5x - 20|}{x-4}
$$





---



### One Sided Limits

[**Video:  Solutions **](https://youtu.be/oDrHcYTWgqI?si=6cGoUKOEi0Z1i-Hb&t=100)


1. A function \( f(x) \) is said to have a **jump discontinuity** at \( x = a \) if:  

   a. \( \displaystyle \lim_{x \to a^-} f(x) \) exists.  
   b. \( \displaystyle \lim_{x \to a^+} f(x) \) exists.  
   c. The left and right limits are **not equal**.

Let  

\[
f(x) =
\begin{cases}
6x-6, & \text{if } x < 8, \\
\frac{2}{x+6}, & \text{if } x \ge 8
\end{cases}
\]

Show that \( f(x) \) has a jump discontinuity at \( x = 8 \).

2. For what value of the constant \( c \) is the function \( f \) continuous on \( (-\infty, \infty) \) where  

\[
f(x) =
\begin{cases}
x^2 - c & \text{if } t < 6, \\
cx + 9, & \text{if } t \ge 6
\end{cases}
\]

Find \( c \).


3. Given the function below, determine if the function is continuous at the point \( x = -4 \). If not, indicate why.

\[
f(x) =
\begin{cases}
2x+9, & \text{if } x < -4, \\
-2x - 7, & \text{if } x > -4
\end{cases}
\]

4. Given the function below, determine if the function is continuous at the point \( x = -1\). If not, indicate why.

\[
f(x) = \ln \|x+1\|
\]



---




### One Sided Limits

[**Video:  Solutions **](https://www.youtube.com/watch?v=PNcd55vzKcI)


1. Let  

$$
f(x) =
\begin{cases}
-\frac{12}{x+6}, & \text{if } x \le 0 \\
\frac{14}{x-7}, & \text{if } x > 0
\end{cases}
$$

Compute the quantities below. 
   a. 
$$
\lim_{x \to -3^-} f(x) = \quad
$$
   b. 
$$
\lim_{x \to -3^+} f(x) = \quad
$$
   c.   
$$
f(-3) = \quad
$$


2. Evaluate the limit:
$$
\lim_{x \to \infty} \frac{5 + 5x}{8 - 4x}
$$


3. Evaluate the limit:

$$
\lim_{x \to \infty} \frac{8x + 7}{6x^2 - 7x + 7}
$$

4. Evaluate the limit:

$$
\lim_{x \to \infty} \frac{\sqrt{9 + 2x^2}}{7 + 6x}
$$

5. Evaluate:

$$
\lim_{t \to \infty} \frac{-3t - 9}{\sqrt{t^2 -4t + 5}}
$$

6. Evaluate the following limits:

   a.  
$$
\lim_{x \to \infty}\frac{8}{e^x - 9}
$$

   b.  
$$
\lim_{x \to -\infty} \frac{8}{e^x - 9}
$$





---



## Practice Problems


Practice the techniques discussed in class and in the online videos by solving the following examples.

1. Estimate $\lim_{x \to 2} e^{x-2}$.

2. Estimate $\lim_{x \to 0^+} \frac{-2}{x^2}$.

3. Evaluate $\lim_{x \to 0} \frac{x-3}{x-1}$.

4. Evaluate $\lim_{x \to 2} \frac{x^{2} - 8x + 12}{x-2}$.

5. Evaluate $\lim_{x \to -2} \frac{x+2}{x^{2} - 4}$.

6. Evaluate $\lim_{x \to \infty} e^{-x}$.

7. Evaluate $\lim_{x \to \infty} \frac{3x - x^{6} + 2}{3x^{3} + 2x}$.

8. Evaluate $\lim_{x \to -\infty} \frac{1 - 3x}{2x^{2} + 3}$.

9. Find a number $b$ so that 
$$
f(x) = \begin{cases} 
5x - 6 & x \leq 2 \\
-3x + b & x > 2
\end{cases}
$$
is continuous everywhere.

10. Find a number $a$ so that 
$$
f(x) = \begin{cases}
ax - 3 & x \leq 3 \\
x + a & x > 3
\end{cases}
$$
is continuous everywhere.

11. (Applied) The percentage of movie advertising as a share of newspapers’ total advertising revenue from 1995 to 2004 can be approximated by
$$
p(t) = \begin{cases}
-0.07t + 6 & t \leq 4 \\
0.3t + 17 & t > 4
\end{cases}
$$
where $t$ is the time since 1995.

   a. Compute $\lim_{t \to 4^-} p(t)$ and $\lim_{t \to 4^+} p(t)$ and interpret each answer.

   b. Is the function $p$ continuous at $t = 4$? What does the answer tell you about newspaper revenues?
</span>

---





---



## Self Assessment




Time yourself and try to solve the following questions within twenty minutes.

1. Evaluate $\lim_{x \to -1} \frac{4x^2 + 1}{x}$.

2. Evaluate $\lim_{x \to -4} \frac{x + 4}{x^{2} - 16}$.

3. Evaluate $\lim_{x \to \infty} \frac{60 + e^{-x}}{2 - e^{-x}}$.

4. Your friend Fiona claims that the study of limits is silly; all you ever need to do to find the limit as $x$ approaches $a$ is substitute $x = a$. Give two examples that show she is wrong.

5. Find a number $b$ so that 
$$
f(x) = \begin{cases}
2x + 1 & x \leq -3 \\
-x + b & x > -3
\end{cases}
$$
is continuous everywhere.




---




## Lesson Checklist 


This checklist is designed to help you keep track of what you need to work on. The main goal is to be aware of what you need to focus more attention on. Place an $X$ in the appropriate box beside the skill below. 
\bigskip

\noindent
\begin{align*}
&\textbf{Developing (D):} &&\textrm{You still need to work on this skill.}\\
&\textbf{Consistent (CON):} &&\textrm{You use the skill correctly most of the time.}\\
&\textbf{Competent (COM):} &&\textrm{You show mastery of the skill.} 
\end{align*}


| Skill                                                          | D  | CON | COM |
|----------------------------------------------------------------|----|-----|-----|
| Estimate a limit using a table of values or by direct substitution. |    |     |     |
| Use factoring to evaluate a limit.                             |    |     |     |
| Use dominant terms analysis to evaluate limits approaching infinity. |    |     |     |
| Find a parameter to make a piecewise function continuous.      |    |     |     |
| Solve applied problems involving limits.                       |    |     |     |


---



















# The Limit Definition of the Derivative






## Lecture Content

[**Video:  Average Rate of Change**](https://youtu.be/0lMCRhUcCtA)

[**Video:  Introduction to the Derivative**](https://youtu.be/yuNhYRZREwI)

[**Video:  Powers, Sums and Constant Multiples**](https://youtu.be/cfP_04TuJc8)



---



## Lecture Notes

### Limit Definition of the Derivative

The derivative of a function \( f(x) \) at a point \( x = a \) is defined as:

\[
f'(a) = \lim_{h \to 0} \frac{f(a + h) - f(a)}{h}
\]

Alternatively, using \( x \):

\[
f'(x) = \lim_{x \to a} \frac{f(x) - f(a)}{x - a}
\]

This represents the **instantaneous rate of change** or the **slope of the tangent line** to the curve at \( x = a \).


---


### Derivative Rules for Common Functions

#### Polynomials
\[
\frac{d}{dx}(x^n) = nx^{n-1}
\]


---


#### Exponential Functions
\[
\frac{d}{dx}(e^x) = e^x, \quad \frac{d}{dx}(a^x) = a^x \ln a
\]


---


#### Logarithmic Functions
\[
\frac{d}{dx}(\ln x) = \frac{1}{x}, \quad \frac{d}{dx}(\log_a x) = \frac{1}{x \ln a}
\]


---


#### Trigonometric Functions
\[
\frac{d}{dx}(\sin x) = \cos x, \quad \frac{d}{dx}(\cos x) = -\sin x
\]
\[
\frac{d}{dx}(\tan x) = \sec^2 x
\]


---


### Sum, Difference, and Constant Multiple Rules

- **Sum Rule:**  
\[
\frac{d}{dx}[f(x) + g(x)] = f'(x) + g'(x)
\]

- **Difference Rule:**  
\[
\frac{d}{dx}[f(x) - g(x)] = f'(x) - g'(x)
\]

- **Constant Multiple Rule:**  
\[
\frac{d}{dx}[c \cdot f(x)] = c \cdot f'(x)
\]

These rules allow for simple combination and manipulation of derivative calculations.


---


### Tangent Line to a Function at a Point

To find the equation of the tangent line to \( f(x) \) at \( x = a \):

1. **Find \( f(a) \)** – the point of tangency: \( (a, f(a)) \).
2. **Compute \( f'(a) \)** – the slope of the tangent line.
3. **Use point-slope form:**

\[
y - f(a) = f'(a)(x - a)
\]

This gives the linear approximation of the function near \( x = a \).

**Example:**

Find the tangent line to \( f(x) = x^2 \) at \( x = 3 \):

- \( f(3) = 9 \)
- \( f'(x) = 2x \Rightarrow f'(3) = 6 \)
- Equation: \( y - 9 = 6(x - 3) \Rightarrow y = 6x - 9 \)




---



## Examples

### Average ROC

[**Video:  Solutions **](https://www.youtube.com/watch?v=KeHgAUW8RpE)


1. The following chart shows *"living wage"* jobs in Rochester per 1000 working-age adults over a 5-year period.

| Year | 1997 | 1998 | 1999 | 2000 | 2001 |
|------|------|------|------|------|------|
| Jobs | 660  | 735  | 795  | 840  | 870  |

   a. What is the average rate of change in the number of living wage jobs from **1997** to **1999**? 

   b. What is the average rate of change in the number of living wage jobs from **1999** to **2001**? 
   

2. Find the slope of the line through the points $(-3,0)$ and $(1,3)$.

3. An object is moving along a vertical line. Its vertical position is given by:
$$
L(t) = t^3  -4t^2 - 4t + 10
$$
where distance is measured in meters and time in seconds.  Find the approximate average velocity (accurate to at least 3 decimal places) in each time interval:

   a.Between $t_1 = 2 \ \text{s}$ and $t_2 = 7 \ \text{s}$
   
   b. Between $t_1 = 1 \ \text{s}$ and $t_2 = 7 \ \text{s}$
   
   c. Between $t_1 = 4 \ \text{s}$ and $t_2 = 5 \ \text{s}$
   
4. Let $f(x) = 3 \ln(x)$.  Using the formula: 
$$
m = \frac{f(x_2) - f(x_1)}{x_2 - x_1}
$$
with $x_1 = 7$, find the slope of the secant line for each $x_2$ value listed below:

| $x_2$  | $m$ (slope) |
|--------|------------|
| 12      | $\_\_\_\_\_\_$ |
| 9      | $\_\_\_\_\_\_$ |
| 8      | $\_\_\_\_\_\_$ |
| 7.1    | $\_\_\_\_\_\_$ |
| 7.01   | $\_\_\_\_\_\_$ |



---



### Average ROC

[**Video:  Solutions **](https://www.youtube.com/watch?v=MirSdnrFyao)


1. Suppose that  
\( f(x) = -2x^2 - 2x -4 \). Find the derivative of $f$.

2. Let \( f(x) = 12x^2 - 7x + 3 \).  Simplify the difference quotient:
\[
\frac{f(3 + h) - f(3)}{h}
\]


3. If \( f(x) = -3x^2 + 7x + 7 \), find the instantaneous rate of change of \( f(x) \) at \( x = -3 \).


4. If \( f(x) = \frac{4}{x^2} \), find \( f'(3) \).




---





### Derivatives using Limits

[**Video:  Derivatives **](hhttps://youtu.be/ebtvadojDKc?si=Gzk1ZKkwJS-RmFwL&t=33)


1. Given \(f(x)=5 x^{2}+ 12x+7\), find \(f'(x)\) using the limit definition of the derivative.


2. Let
\[
f(x)=3x^{3} + 7x - 4.
\]
Use the limit definition of the derivative to calculate the derivative and second derivative of \(f\).


3. For what values of \(A\) and \(B\) is the function \(f(x)\) differentiable at \(x=-2\)?

\[
f(x)=
\begin{cases}
Ax+B, & x< 2,\\[6pt]
x^{2}+x, & x\ge 2.
\end{cases}
\]


4. Use the limit definition to find the derivative of
\[
f(x)=\sqrt{x+5}.
\]



---




### Basic Derivative Rules

[**Video: Solutions**](https://www.youtube.com/watch?v=491OOM9Bgxc)


1. Find the derivative of the function $f(x) = 7x^{3} - 10x^{7}$.

2. If \(f(x) = \frac{4}{x^2\), find the value of the derivative at \(x=1\):

3. Find the derivative of the function: $f(x) = 2\sqrt{x} - \frac{9}{x^9}$.

4.Given $f(x) = 4x\sqrt{x} + \frac{5}{5x^{2}\sqrt{x}}$, find $f'(x)$ and then compute $f'(1) =$

5. Let $h(t) = 5t^{3.2} - 8t^{-3.2}$. Find $h'(t)$, $h'(4)$, $h''(t)$, and $h''(4)$

6. If
\[
g(t) = -2t^{4} + 5t^{2} + 5,
\]
find the following values:
\[
\begin{aligned}
g(0) &= \underline{\qquad\qquad} \\
g'(0) &= \underline{\qquad\qquad} \\
g''(0) &= \underline{\qquad\qquad} \\
g'''(0) &= \underline{\qquad\qquad} \\
g^{(4)}(0) &= \underline{\qquad\qquad} \\
g^{(5)}(0) &= \underline{\qquad\qquad}
\end{aligned}
\]




---



## Practice Problems



Practice the techniques discussed in class and in the online videos by solving the following examples.

1. Use the limit definition of a derivative to find the derivative of $f(x) = x^{2} - 3$.

2. Use the limit definition of a derivative to find the derivative of $f(x) = \frac{2}{x^{2}}$.

3. Differentiate the following functions:

   a. $f(x) = x^{5} + 2x - 2$.

   b. $f(x) = |x| + \frac{1}{x} + \ln(x)$.

   c. $f(x) = \log_{3} x$.

   d. $g(x) = 5^{x} + e^{x}$.

   e. $y = 4x^{-1} - 2|x| - 10 + \sin(x)$.

   f. $f(x) = 3x^{3} - 2x^{2} + \sqrt{x}$.

   g. $f(x) = 5 \sin(x) + \ln(x)$.

   h. $f(x) = \cos(x) - 3 \sin(x) + e^{x}$.

4. Find the equation of the tangent line to the equation below at the indicated point:

   a. $f(x) = x + \frac{1}{x}$ at $x=2$.

   b. $f(x) = \frac{1}{x^{2}}$ at $x=1$.

   c. $f(x) = \sqrt{x}$ at $x=4$.

5. (Applied) Company C’s profits are given by $P(0) = \$1$ million and $P'(0) = \$0.5$ million/month. Company D’s profits are given by $P(0) = \$0.5$ million and $P'(0) = \$1$ million/month. In which company would you rather invest? Why?

6. (Applied) The cost of producing $x$ teddy bears per day at the Cuddly Companion Co. is calculated by their marketing staff to be given by the formula
$$
C(x) = 100 + 40x - 0.001 x^{2}.
$$

   a. Find the marginal cost function and use it to estimate how fast the cost is going up at a production level of 100 teddy bears. Compare this with the exact cost of producing the 101st teddy bear.

   b. Find the average cost function $\overline{C}$, and evaluate $\overline{C}(100)$. What does the answer tell you?
</span>

---

## Self Assessment


Time yourself and try to solve the following questions within twenty minutes.

1. Use the limit definition of a derivative to find the derivative of $f(x) = x - 2x^{3}$.

2. Differentiate the following functions:

   a. $f(x) = 2x^{4} + 3x^{3} - 1$.

   b. $y = x^{2} + 3|x| - \cos(x)$.

   c. $f(x) = x^{2} - 3\sqrt{x} + 5$.

   d. $f(x) = 5 \sin(x) + \ln(x)$.

   e. $f(x) = \cos(x) - 3 \sin(x) + e^{x}$.

3. Find the equation of the tangent line to $f(x) = x^{2} + \cos(x)$ at $x=0$.

4. Daily oil production by Pemex, Mexico’s national oil company, can be approximated by
$$
P(t) = -0.022 t^{2} + 0.2 t + 2.9 \text{ million barrels } \quad (1 \leq t \leq 9),
$$
where $t$ is time in years since the start of 2000. Find the derivative function $dP/dt$. At what rate was oil production changing at the start of 2004 ($t=4$)?

5. A car wash firm calculates that its daily profit (in dollars) depends on the number $n$ of workers it employs according to the formula
$$
P = 400 n - 0.5 n^{2}.
$$
Calculate the marginal product at an employment level of 50 workers, and interpret the result.


---


## Lesson Checklist 


This checklist is designed to help you keep track of what you need to work on. The main goal is to be aware of what you need to focus more attention on. Place an $X$ in the appropriate box beside the skill below. 
\bigskip

\noindent
\begin{align*}
&\textbf{Developing (D):} &&\textrm{You still need to work on this skill.}\\
&\textbf{Consistent (CON):} &&\textrm{You use the skill correctly most of the time.}\\
&\textbf{Competent (COM):} &&\textrm{You show mastery of the skill.} 
\end{align*}



| Skill                                                        | D  | CON | COM |
|--------------------------------------------------------------|----|-----|-----|
| Use the limit definition to find the derivative of a function.|    |     |     |
| Find the derivatives of polynomial and absolute value functions.|    |     |     |
| Find the derivatives of exponential and logarithmic functions.|    |     |     |
| Find the derivative of trigonometric functions.               |    |     |     |
| Determine the equation of a tangent line at a given point.    |    |     |     |
| Solve applied problems involving derivatives.                 |    |     |     |



---



# (PART) Derivative Formulas and Applications {-}




# The Product and Quotient Rules



## Lecture Content

[**Video:  The Product and Quotient Rules**](https://youtu.be/hyq3m2K10fI)



---


## Lecture Notes



### Product Rule for Two Functions

If \( f(x) \) and \( g(x) \) are differentiable, then the derivative of their product is:

\[
\frac{d}{dx}[f(x)g(x)] = f'(x)g(x) + f(x)g'(x)
\]

**Mnemonic:** "First times the derivative of the second, plus second times the derivative of the first."

---

### Product Rule for Three Functions

If \( f(x), g(x), h(x) \) are differentiable, then the derivative of their product is:

\[
\frac{d}{dx}[f(x)g(x)h(x)] = f'(x)g(x)h(x) + f(x)g'(x)h(x) + f(x)g(x)h'(x)
\]

**Tip:** Differentiate one function at a time, keeping the other two unchanged, and sum the results.

---

### Quotient Rule

If \( f(x) \) and \( g(x) \) are differentiable and \( g(x) \ne 0 \), then:

\[
\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2}
\]

**Mnemonic:** "Low D high minus high D low, over low squared."

---


## Exercises



### Applications of Differentiation

[**Video: Solutions**](https://www.youtube.com/watch?v=7XOzaaMWwbM)


1. Suppose a product's revenue function is given by
\[
R(q) = -3q^{2} + 800q.
\]
Find an expression for the marginal revenue function and simplify it.

2. The R.C. Helliot Advertising Company finds that its profits on day \(t\) of an advertising campaign is given by
\[
P(t) = -2.53t^{2} + 65t + 37,000,
\]
where \(P(t)\) is the profit in dollars for day \(t\).

   a. Find a simplified expression for the marginal profit function. Use the proper variable \(t\).
   b. What is the exact rate of change of profits when 44 days have passed in the campaign? 
   c. What are the total profits on day 44 of the campaign?

3. A simple item to be sold at a small specialty store has \$200 in fixed costs and \$8 per item in variable costs. The owners assume they can sell 8 items if they charge \$28 each, and 5 items at a price of \$55 each. (Assume that demand is a linear function.)

For how many units sold/produced is the marginal profit -124 dollars per item? (Use derivatives to find your answer.)

4. In 1997, researchers at Texas A&M University estimated the operating costs of cotton gin plants of various sizes. A quadratic model of cost (in thousands of dollars) for the largest plants was found to be
\[
C(q) = 0.021q^{2} + 21.2q + 387,
\]
where \(q\) is the annual quantity of bales (in thousands) produced by the plant. Revenue was estimated at \$68 per bale of cotton.

Find the following:

   a. $MC(q)$
   b. The Marginal Revenue function
   c. The Marginal Profit function
   d. The Marginal Profit for \(q = 262\) thousand units
   e. What are the proper units for the answer to Part D?




---




### The Product and Quotient Rules

[**Video: Solutions**](https://www.youtube.com/watch?v=g9knx4EPqEc)


1. If $f(t) = (t^2 + 2t + 5)(6t^2 + 5)$, find \(f'(t)\) and \(f'(4)\).

2. Find the derivative of $\cot(x) \cdot \frac{1}{x^3} (5x + 2)$.

3. Find the derivative of $f(x) = 16 \sqrt{x} \left(3x^4 - 3\right) \left(6x^8 + 7x\right)$.

4. If $f(x) = (3x^2 + 6x -5) \tan(x)$, find $f'(x)$.

5. If $f(x) = \frac{\sqrt{x} - 4}{\sqrt{x} + 4}$, find $f'(x)$ and $f'(3)$.

6. If $f(x) = \frac{7x + 4}{7x + 5}$, find $f'(x)$ and $f'(5)$.

7. If $f(x) = \frac{7 - x^2}{2 + x^2}$, find $f'(x)$

8. Find the first and second derivatives of $f(x) = \frac{\sin(x)}{x^2}$.



---



## Practice Problems


Practice the techniques discussed in class and in the online videos by solving the following examples.

1. Differentiate:

   a. $y = x^{2} \left( 4x^{3} + \frac{2}{x} - 1 \right)$.

   b. $f(x) = x \ln x$.

   c. $f(x) = (2x^{0.5} - x^{2})^{2}$.

2. Differentiate:

   a. $y = \sqrt{x} \left( 4x^{-1} - 2|x| - 10 \right)$.

   b. $f(x) = x (x^{2} - 3)(2x^{2} + 1)$.

   c. $y = \left( \frac{x^{1.2}}{7} + \frac{2}{x^{2.1}} \right) \cos(x)$.

3. Differentiate:

   a. $f(x) = \frac{2x + 4}{3x - 1}$.

   b. $y = \tan(x) = \frac{\sin(x)}{\cos(x)}$.

   c. $f(x) = \frac{e^{2x}}{\ln(2x) + 2x}$.

4. Differentiate:

   a. $f(x) = \frac{(x-3)(x-2)(x-1)}{x+5}$.

   b. $y = \frac{\sqrt{x} - 1}{\sqrt{x} + 1}$.

   c. $y = \frac{(x+1)(x+2)}{(x-3)(x-2)(x-1)}$.

5. (Applied) The monthly sales of Sunny Electronics’ new iSun walkman is given by
$$
q(t) = 2000 t - 100 t^{2}
$$
units per month, $t$ months after its introduction. The price Sunny charges is
$$
p(t) = 100 - t^{2}
$$
dollars per iSun, $t$ months after introduction. Find the rate of change of monthly sales, the rate of change of the price, and the rate of change of monthly revenue six months after the introduction of the iSun. Interpret your answers.


---


## Self Assessment

Time yourself and try to solve the following questions within twenty minutes. 

1. Differentiate: $f\left(x\right) = x^2\left(2x + 3\right)\left(7x + 2\right)$
2. Differentiate: $y =\sin(x)\left(x^2 - 1\right)$
3. Differentiate: $y =\dfrac{x}{\sqrt{x} + |x|}$
4. Differentiate: $y =\dfrac{\frac{1}{x}-1}{\sin(x)}$
5. Thoroughbred Bus Company finds that its monthly costs for one particular year were given by $C(t) = 100 + t^2$ dollars after $t$ months. After $ t$ months, the company had $P(t) = 1,000 + t^2$ passengers per month. How fast is its cost per passenger changing after 6 months? 




---



## Lesson Checklist 

This checklist is designed to help you keep track of what you need to work on. The main goal is to be aware of what you need to focus more attention on. Place an $X$ in the appropriate box beside the skill below. 
\bigskip

\noindent
\begin{align*}
&\textbf{Developing (D):} &&\textrm{You still need to work on this skill.}\\
&\textbf{Consistent (CON):} &&\textrm{You use the skill correctly most of the time.}\\
&\textbf{Competent (COM):} &&\textrm{You show mastery of the skill.} 
\end{align*}



| Skill                                                        | D  | CON | COM |
|--------------------------------------------------------------|----|-----|-----|
| Find the derivatives using the product rule.|    |     |     |
| Find the derivatives using the quotient rule.|    |     |     |











---







# The Chain Rule and L'Hopital's Rule





## Lecture Content

[**Video:  The Chain Rule**](https://youtu.be/nU0kXDFSt_A)

[**Video:  L'Hospital's Rule**](https://youtu.be/sZv3uLLOwFc)


---



## Lecture Notes

### Derivative of a Composition of Functions (Chain Rule)

If a function is defined as a composition \( y = f(g(x)) \), then the derivative is given by the **chain rule**:

\[
\frac{dy}{dx} = f'(g(x)) \cdot g'(x)
\]

**In Leibniz notation:**
If \( y = f(u) \) and \( u = g(x) \), then:

\[
\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}
\]

**Example:**
\[
\text{If } y = \sin(x^2), \text{ then } \frac{dy}{dx} = \cos(x^2) \cdot 2x
\]

---

### L'Hôpital's Rule

L’Hôpital’s Rule provides a method to evaluate limits of indeterminate forms \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \).

#### Statement of the Rule

Let \( f(x) \) and \( g(x) \) be differentiable near \( a \), and suppose:

- \( \lim_{x \to a} f(x) = \lim_{x \to a} g(x) = 0 \), or
- \( \lim_{x \to a} f(x) = \lim_{x \to a} g(x) = \infty \)

and

- \( g'(x) \ne 0 \) near \( a \), and
- \( \lim_{x \to a} \frac{f'(x)}{g'(x)} \) exists (or is \( \pm\infty \)),

then:

\[
\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}
\]

**Note:** The rule also applies as \( x \to \infty \) or \( x \to -\infty \).

---

**Example:**
\[
\lim_{x \to 0} \frac{\sin x}{x} = \frac{0}{0} \Rightarrow \text{Apply L'Hôpital's Rule:} \quad \lim_{x \to 0} \frac{\cos x}{1} = 1
\]


---


## Examples

### The Chain Rule

[**Video: Solutions**](https://www.youtube.com/watch?v=IseqLbi29q0)


1. Use the chain rule to find the derivative of $10 \sqrt{10x^9 + 3x^5}$.

2. Use the chain rule to find the derivative of $\left(2(10x^6 - 9x^{4})\right)^{16}$.

3. If $f(x) = \cos(4x + 6)$, find $f'(x)$ and f'(5)$.

4. Find the derivative of $-7 \sin^2(-6x^9)$.

5. Let $f(x) = 5 \cos(\cos x)$. Find $f'(x)$

6. Let $f(x) = (7x^2 - 4)^4 (-9x^2 + 4)^{14}$. Find $f'(x)$.

7. Let $g(s) = (4s - 4)^8$. Find $g'(s)$, $g'(5)$, $g''(s)$ and $g''(5)$.


---



### Derivatives of Logs and Exponentials

[**Video: Solutions**](https://www.youtube.com/watch?v=LU037z7puWw)


1. Let $f(t) = 7t^4 - 7t + 6e^t$.  Find $f'(t)$.

2. Find the derivative of $f(x) = x^5 e^{6x}$. Find $f'(x)$.

3. Let $f(x) = 5x^7 \ln(x) - \frac{4}{3}x^3$.  Find $f'(x)$.

4. Let $f(x) = -4x^2 \ln x$.  Find $f'(x)$ and $f'(e^2)

5. Differentiate $g(x) = \frac{e^{8x}}{x^2}$.

6. If $f(y) = e^{7y} - e^{-7y}$, find $f'(y)$.

7. Let $f(x) = \ln(x^2 + 10x + 29)$.  Find $f'(x)$.

8. Let $f(x) = 4 \ln(\sin x)$. Find $f''(x)$.




---




## Practice Problems


Practice the techniques discussed in class and in the online videos by solving the following examples.

1. Differentiate:
   a. $f(x) = (3x - 1)^{10}$
   
   b. $y = (1 - x)^{-1}$
   
   c. $v(x) = 3^{2x + 1} + e^{3x + 1}$

2. Differentiate:
   a. $f(x) = (x^2 + 1)^5 \ln x$
   
   b. $f(x) = \frac{1}{(x + 1)^2}$
   
   c. $g(x) = \ln(x^2 + 3)$

3. Differentiate:
   a. $h(x) = \ln \left(\frac{9x}{4x - 2}\right)$
   
   b. $h(x) = 2[(x + 1)(x^2 - 1)]^{-1/2}$
   
   c. $h(x) = 3^{x^2 - x}$


4. Verify L'Hopital's Rule can be applied, and then use it to evaluate the limit. Sometimes you may have to apply the rule more than once.

   a. $$\lim_{x \to 1} \frac{x^2 - 2x + 1}{x^2 - x}$$
   
   b. $$\lim_{x \to -2} \frac{x^3 + 8}{x^2 + 3x + 2}$$
   
   c. $$\lim_{x \to \infty} \frac{6x^2 - 5x + 1}{3x^2 - 9}$$
   
   d. $$\lim_{x \to \infty} \frac{x^2}{e^x}$$


5. (Applied) The average price of a two-bedroom apartment in downtown New York City during the real estate boom from 1994 to 2004 can be approximated by

$$
p(t) = 0.33 e^{0.16 t} \text{ million dollars } \quad (0 \leq t \leq 10),
$$

where $t$ is time in years ($t = 0$ represents 1994). What was the average price of a two-bedroom apartment in downtown New York City in 2003, and how fast was it increasing?

6. Find the equation of the tangent line to $$y = \ln \sqrt{2x^2 + 1}$$ at $x=1$.


---



## Self Assessment


Time yourself and try to solve the following questions within twenty minutes.

1. Differentiate: $$r(x) = \left(\sqrt{x+1} + \sqrt{x}\right)^3$$

2. Differentiate: $$r(x) = \left(e^{2x^2}\right)^3$$

3. Determine the limit: $$\lim_{x \to -\infty} \frac{4x^3 + x^2 - 2x}{x^2 - 7}$$

4. Determine the limit: $$\lim_{x \to \frac{1}{2}} \frac{\sin(2x - 1)}{2x - 1}$$

5. The total spent on research and development by the federal government in the United States during 1995–2007 can be approximated by

$$
S(t) = 7.4 \ln t + 3 \text{ billion dollars} \quad (5 \leq t \leq 19)
$$

where $t$ is the year since 1990. What was the total spent in 2005 $(t = 15)$ and how fast was it increasing?


---



## Lesson Checklist 


This checklist is designed to help you keep track of what you need to work on. The main goal is to be aware of what you need to focus more attention on. Place an $X$ in the appropriate box beside the skill below. 
\bigskip

\noindent
\begin{align*}
&\textbf{Developing (D):} &&\textrm{You still need to work on this skill.}\\
&\textbf{Consistent (CON):} &&\textrm{You use the skill correctly most of the time.}\\
&\textbf{Competent (COM):} &&\textrm{You show mastery of the skill.} 
\end{align*}



| Skill                                      | D  | CON | COM |
|--------------------------------------------|----|-----|-----|
| Find the derivative using the chain rule. |    |     |     |
| Apply L'Hopital's Rule to an indeterminate limit. |    |     |     |
| Solve applied problems using advanced derivative rules. |    |     |     |








---






# Implicit Differentiation






## Lecture Content


[**Video:  Higher Order Derivatives**](https://youtu.be/wgX5LHbb6pE)

[**Video:  Derivatives of Logarithmic and Exponential Functions**](https://youtu.be/UeazZ3n_n20)

[**Video:  Implicit Differentiation**](https://youtu.be/s6_iOkjPnD4)


---



## Lecture Notes

### Higher-Order Derivatives

Higher-order derivatives are the successive derivatives of a function.

- **First derivative:** \( f'(x) \) — represents the rate of change or slope.
- **Second derivative:** \( f''(x) \) — measures the rate of change of the first derivative (i.e., concavity or acceleration).
- **Third and beyond:** \( f^{(3)}(x), f^{(4)}(x), \ldots \) are called third, fourth, etc., derivatives.

**Notation:**

- \( f^{(n)}(x) \) is the \( n \)th derivative.
- In physics, these can represent velocity, acceleration, jerk, etc.

**Example:**
If \( f(x) = x^3 \):

- \( f'(x) = 3x^2 \)
- \( f''(x) = 6x \)
- \( f^{(3)}(x) = 6 \)

---

### Implicit Differentiation

Implicit differentiation is used when a function is not given in the form \( y = f(x) \), but instead in an implicit form, like an equation involving both \( x \) and \( y \).

#### Steps:

1. **Differentiate both sides** of the equation with respect to \( x \).  
   - Treat \( y \) as a function of \( x \) (i.e., apply the chain rule to any term involving \( y \)).
2. **Apply the chain rule:**  
   - When differentiating terms with \( y \), include \( \frac{dy}{dx} \) (or \( y' \)).
3. **Solve for \( \frac{dy}{dx} \)** (i.e., isolate the derivative on one side).

---

**Example:**
Given \( x^2 + y^2 = 25 \), find \( \frac{dy}{dx} \):

1. Differentiate both sides:
\[
\frac{d}{dx}(x^2) + \frac{d}{dx}(y^2) = \frac{d}{dx}(25)
\Rightarrow 2x + 2y \frac{dy}{dx} = 0
\]

2. Solve for \( \frac{dy}{dx} \):
\[
2y \frac{dy}{dx} = -2x \Rightarrow \frac{dy}{dx} = -\frac{x}{y}
\]

---



## Examples


[**Video:  Solutions**](https://www.youtube.com/watch?v=bEevIpR4W1I)

1. Use implicit differentiation to determine \(\frac{dy}{dx}\) given the equation $x^5 + y^6 = -7$.

2. Given $4x^2 + 2x + xy = 2$, and \(y(2) = -9\), find \(y'(2)\) by implicit differentiation.

3. Find the slope of the tangent line to the curve $4x^2 + 2xy - 4y^3 = 6$ at the point \((1, -1)\).



---



## Practice Problems

Practice the techniques discussed in class and in the online videos by solving the following examples.

1. Find the first $n$ derivatives of $$f(x) = -3x^3 + 4x.$$

2. Find $dy/dx$ given $$x^2 y - y^2 = 4.$$

3. Find $dy/dx$ given $$3xy - \frac{y}{x} = 2.$$

4. Find $dy/dx$ given $$y \ln x + y = 2.$$

5. Find $dy/dx$ given $$x e^y = y e^x.$$

6. Differentiate $$y = \frac{(x+1)(x+2)(x+3)(x+4)}{x(x-1)(x-2)(x-3)}.$$

7. Find the tangent line to $$e^{-xy} + 2x = 1$$ at $x = -1$.

8. (Applied) An employment research company estimates that the value of a recent MBA graduate to an accounting company is 
$$
V = 3e^2 + 5g^3,
$$
where $V$ is the value of the graduate, $e$ is the number of years of prior business experience, and $g$ is the graduate school grade point average. If $V$ is fixed at 200, find $de/dg$ when $g = 3.0$ and interpret the result.

9. (Applied) Use logarithmic differentiation to give a proof of the quotient rule.




---



## Self Assessment

Time yourself and try to solve the following questions within twenty minutes.

1. Find the first $n$ derivatives of $$f(x) = (-2x + 1)^3.$$

2. Find $dy/dx$ given $$\ln(y^2 - y) + x = y.$$

3. Find $dy/dx$ given $$e^x y = 1.$$

4. Differentiate $$y = \frac{(3x + 1)^2}{4x(2x - 1)^3}.$$

5. Find the tangent line to $$4x^2 + 2y^2 = 12$$ at $(1, -2)$.


---


## Lesson Checklist 


This checklist is designed to help you keep track of what you need to work on. The main goal is to be aware of what you need to focus more attention on. Place an $X$ in the appropriate box beside the skill below. 
\bigskip

\noindent
\begin{align*}
&\textbf{Developing (D):} &&\textrm{You still need to work on this skill.}\\
&\textbf{Consistent (CON):} &&\textrm{You use the skill correctly most of the time.}\\
&\textbf{Competent (COM):} &&\textrm{You show mastery of the skill.} 
\end{align*}



| Skill                                      | D  | CON | COM |
|--------------------------------------------|----|-----|-----|
| Find higher order derivatives of a function. |    |     |     |
| Use implicit differentiation to find the derivative. |    |     |     |
| Use logarithmic differentiation to find the derivative. |    |     |     |










---






# Graphical Analysis



## Lecture Content

[**Video:  Analyzing Graphs**](https://youtu.be/f30X6LIzYpM)

[**Video:  Maxima and Minima**](https://youtu.be/mM0A9Kv_wYw)



---


## Lecture Notes


### Finding Critical Numbers

Critical numbers occur where the first or second derivative is zero or undefined.

#### Steps for the First Derivative \( f'(x) \):

1. Find \( f'(x) \).
2. Solve \( f'(x) = 0 \) or where \( f'(x) \) is undefined.
3. The values of \( x \) where this occurs are **critical numbers** (provided \( x \) is in the domain of \( f \)).

#### Steps for the Second Derivative \( f''(x) \):

1. Find \( f''(x) \).
2. Solve \( f''(x) = 0 \) or where \( f''(x) \) is undefined.
3. These are **possible inflection points** (to be confirmed by a sign change).

---

### Classifying Maximums, Minimums, and Inflection Points

#### First Derivative Test (for relative extrema):

- If \( f'(x) \) changes from positive to negative at \( x = c \), then \( f(c) \) is a **local maximum**.
- If \( f'(x) \) changes from negative to positive at \( x = c \), then \( f(c) \) is a **local minimum**.

#### Second Derivative Test (for local extrema):

- If \( f'(c) = 0 \) and \( f''(c) > 0 \), then \( f(c) \) is a **local minimum**.
- If \( f'(c) = 0 \) and \( f''(c) < 0 \), then \( f(c) \) is a **local maximum**.
- If \( f''(c) = 0 \), the test is inconclusive.

#### Inflection Points:

- A point \( x = c \) is an **inflection point** if:
  - \( f''(x) \) changes sign at \( x = c \), and
  - \( f \) is continuous at \( x = c \)

---

### Extreme Value Theorem (EVT)

The **Extreme Value Theorem** states:

> If a function \( f(x) \) is continuous on a closed interval \([a, b]\), then \( f(x) \) has both an absolute maximum and an absolute minimum on that interval.

#### To find the absolute extrema:

1. Find critical numbers of \( f(x) \) in the interval \([a, b]\).
2. Evaluate \( f(x) \) at:
   - Each critical number in \([a, b]\)
   - The endpoints \( a \) and \( b \)
3. Compare all function values. The largest is the **absolute maximum**, and the smallest is the **absolute minimum**.

---

### Intervals of Increase, Decrease, and Concavity

#### Increasing/Decreasing Intervals:

1. Find \( f'(x) \) and critical points.
2. Make a sign chart for \( f'(x) \).
3. If \( f'(x) > 0 \) on an interval, then \( f(x) \) is **increasing** there.
4. If \( f'(x) < 0 \) on an interval, then \( f(x) \) is **decreasing** there.

#### Concavity and Inflection Points:

1. Find \( f''(x) \) and solve \( f''(x) = 0 \).
2. Make a sign chart for \( f''(x) \).
3. If \( f''(x) > 0 \), then \( f(x) \) is **concave up**.
4. If \( f''(x) < 0 \), then \( f(x) \) is **concave down**.
5. Points where concavity changes are **inflection points**.

---

## Examples

### Maximum and Minimum Values

[**Video: Solutions**](https://www.youtube.com/watch?v=ItWO-kYqMdE)

1. Given the function $g(x) = 8x^3 + 72x^2 + 192$, find the first derivative, \(g'(x)\).  Notice that \(g'(x) = 0\) when \(x = -2\), i.e., $g'(-2) = 0$.  Use the second derivative test to determine whether there is a local minimum or maximum at \(x = -2\).  Find the second derivative, \(g''(x)\).  Evaluate \(g''(-2)\).  Based on the sign of this number, is the graph of \(g(x)\) concave up or down at \(x = -2\)?  

2. The function $f(x) = 2x^3 - 36x^2 + 210x + 5$ has one local minimum and one local maximum.

3. Consider the function $f(x) = 5 - 7x^2,$ with $-5 \leq x \leq 1$.  Find the absolute maximum and minimum.

4. Consider the function $f(x) = 2x^3 + 12x^2 - 192x + 8$, with $ -12 \leq x \leq 5$.  Find the absolute maximum and minimum.



---



### Applied Problems

[**Video: Solutions**](https://www.youtube.com/watch?v=dwxsl2lewT8)

1. A rectangular storage area is to be constructed along a side of a building. A security fence is required along the remaining three sides of the area. You have 865 ft of fencing available.  What is the maximum storage area?

2. A rancher wants to fence in an area of 1,000,000 square feet in a rectangular field and then divide it in half with a fence down the middle parallel to one side. Find the shortest length of fence that the rancher can use.

3. A box with a square base and open top must have a volume of 143748 $cm^3$. We want to find the dimensions of the box that minimize the amount of material used.

4. A small object is moving along a vertical line. Its location function is $t^3 - 24t^2 + 45t + 3$ with $0 \leq t \leq 11$.
   - At what time(s) is the object's velocity greatest?
   - At what time(s) is the object's speed greatest?

5. A ball is tossed into the air on the planet Bellona. The height of the ball in meters above the ground \(t\) seconds after being thrown is:
\[
s(t) = \frac{7}{2}t^2 - t^3.
\]
   a. Find the velocity and acceleration functions.
   b. Over what interval(s) of time is the ball moving upward? (Give interval notation)
   c. After initially being thrown, when does the ball return to the ground?
   d. What is the velocity of the ball when it returns to the ground?




---




### Increasing and Decreasing

[**Video: Solutions**](https://www.youtube.com/watch?v=kuR_W7wo0ps)

1. Let $f(x) = x^3 + 3x^2 - 240x + 18$.
   a. Find the first derivative \( f'(x) \).
   b. Find the second derivative \( f''(x) \).
   c. Find the interval(s) where \( f \) is increasing (include endpoints).
   d. Find the interval(s) where \( f \) is decreasing (include endpoints).
   e. Find the interval(s) where \( f \) is concave downward (include endpoints).
   f. Find the interval(s) where \( f \) is concave upward (include endpoints).
   
2. Let $f(x) = \frac{x^4 - 32x^2}{7}$.
   a. Find the first derivative \( f'(x) \).
   b. Find the second derivative \( f''(x) \).
   c. Find the interval(s) where \( f \) is increasing (include endpoints).
   d. Find the interval(s) where \( f \) is decreasing (include endpoints).
   e. Find the interval(s) where \( f \) is concave downward (include endpoints).
   f. Find the interval(s) where \( f \) is concave upward (include endpoints).
   

3. Consider the function $f(x) = 5(x-2)^{2/3}$.
   - Find the critical number \( A \).
   - On each interval below, state if \( f(x) \) is increasing or decreasing.
   - On each interval below, state if \( f(x) \) is concave up (type "CU") or concave down (type "CD").

4. Consider the function $f(x) = x^2 e^{9x}$.
   - Find the critical numbers.
   - Determine the intervals of increasing and decreasing.
   

5. Consider the function $f(x) = x^2 e^{8x}$. Find the intervals of concavity.



---



### Curve Sketching

[**Video: Solutions**](https://www.youtube.com/watch?v=SqaIfGD0soU)


1. Consider the function $f(x) = 2x^3 + 6x^2 - 18x - 54.  Find 
   - the first derivative, 
   - all critical values,
   - intervals of increasing and decreasing,
   - maximum and minimum values,
   - the second derivative,
   - inflection points,
   - intervals of concavity.
   
2. Consider the function $f(x) = \frac{x^2 - 1}{x -3}$.  Find
   - the first derivative, 
   - all critical values,
   - intervals of increasing and decreasing,
   - maximum and minimum values,
   - the second derivative,
   - inflection points,
   - intervals of concavity.

3. Let $f(x) = x^3 -6x^2 - 1$.  Find
   - the first derivative, 
   - all critical values,
   - intervals of increasing and decreasing,
   - maximum and minimum values,
   - the second derivative,
   - inflection points,
   - intervals of concavity.

4. Let $f(x) = x^3 + 9x^2 + 27x + 4$.  Find
   - the first derivative, 
   - all critical values,
   - intervals of increasing and decreasing,
   - maximum and minimum values,
   - the second derivative,
   - inflection points,
   - intervals of concavity.



---




## Practice Problems


Practice the techniques discussed in class and in the online videos by solving the following examples.

1. Find the location of all relative and absolute extrema for $$f(x) = 2x^2 - 2x + 3$$ on $[0, 3]$.

2. Find the location of all relative and absolute extrema for $$f(x) = \sqrt{x}(x+1)$$ on $[0, \infty)$.

3. Find the location of all relative and absolute extrema for $$f(x) = \frac{x^2 + 1}{x^2 - 1}$$ on $[-2, 2]$ with $x \neq \pm 1$.

4. Find the location of all relative and absolute extrema for $$f(x) = (x+1)^{2/5}$$ on $[-2, 0]$.

5. Find and classify the critical points and locate all inflection points for $$f(x) = 2x^2 - 2x + 3.$$

6. Find and classify the critical points and locate all inflection points for $$f(x) = -x^3 + 3x.$$

7. Sketch the graph of $$h(x) = -2x^3 - 3x^2 + 36x.$$

8. Sketch the graph of $$f(t) = \frac{t^2 - 1}{t^2 + 1}.$$

9. (Applied) Maximize $$P = xy$$ with $$x + 2y = 40.$$

10. (Applied) The cost of controlling emissions at a firm rises rapidly as the amount of emissions reduced increases. Here is a possible model: 

$$
C(q) = 4,000 + 100 q^2
$$

where $q$ is the reduction in emissions (in pounds of pollutant per day) and $C$ is the daily cost to the firm (in dollars) of this reduction. What level of reduction corresponds to the lowest average cost per pound of pollutant, and what would be the resulting average cost to the nearest dollar?


---



## Self Assessment

Time yourself and try to solve the following questions within twenty minutes.

1. Find the location of all relative and absolute extrema for $$f(x) = 2x^3 + 3x^2$$ on $[-2, \infty)$.

2. Find and classify the critical points and locate all inflection points for $$f(x) = x e^{-x^2}.$$

3. Sketch the graph of $$g(x) = \frac{x^3}{x^2 - 3}.$$

4. Hercules Films is also deciding on the price of the video release of its film *Bride of the Son of Frankenstein*. Again, marketing estimates that at a price of $p$ dollars, it can sell 
$$
q = 200,000 - 10,000 p
$$
copies, but each copy costs \$4 to make. What price will give the greatest profit?

5. Let 
$$
f(x) = \frac{N}{1 + A e^{-k x}}
$$
for constants $N$, $A$, and $k$ ($A$ and $k$ positive). Show that $f$ has a single point of inflection at $$x = \frac{\ln A}{k}.$$



---


## Lesson Checklist 


This checklist is designed to help you keep track of what you need to work on. The main goal is to be aware of what you need to focus more attention on. Place an $X$ in the appropriate box beside the skill below. 
\bigskip

\noindent
\begin{align*}
&\textbf{Developing (D):} &&\textrm{You still need to work on this skill.}\\
&\textbf{Consistent (CON):} &&\textrm{You use the skill correctly most of the time.}\\
&\textbf{Competent (COM):} &&\textrm{You show mastery of the skill.} 
\end{align*}




| Skill                                                        | D  | CON | COM |
|--------------------------------------------------------------|----|-----|-----|
| Find and classify critical numbers of the first derivative.  |    |