Derivatives

Use limits to give mathematical definitions for each of the following derivatives, first with the $h\to 0$ definition of the derivative, and then with the $z\to x$definition:

(a) the derivative of a function f at the point $x=5.$

(b) the derivative of a function f

(c) the right derivative of a function f at the point $x=-2.$

 Find a function $f$ that has the given derivative $f^{\text{'}}$ and value f(c).

$f^{\text{'}}\left(x\right)=\frac{3x^2}{\sqrt{x^3+1}},f\left(2\right)=6$

Find a function f that has the given derivative $f\text{'}$  and value f(c).

$f^{\text{'}}\left(x\right)=8e^{4x}+1,f\left(0\right)=3$

To save up for a car, you take a job working 10 hours a week at the school library. For the first six weeks, the library pays you $\backslash (8.00$ an hour. After that, you earn $\backslash )11.50$ an hour. You put all of the money you earn each week into a savings account. On the day you start to work your savings. account already holds $\$200.00$. Let S(t) be the function that describes the amount in your savings account t weeks after your library job begins.

(a) Find the values of $S\left(3\right), S\left(6\right), S\left(8\right), S\text{'}\left(3\right), S\text{'}\left(6\right) and S\text{'}\left(8\right)$,  if possible, and describe their meanings in practical terms. If it is not possible to find one or more of these values, explain why.

(b) Write an equation for the function $S\left(t\right)$. (Hint: $S\left(t\right)$ will be a piecewise-defined function.) Be sure that your equation correctly produces the values you calculated in part (a).

(c) Sketch a labeled graph of $S\left(t\right)$. By looking at this graph, determine whether $S\left(t\right)$ is continuous and whether $S\left(t\right)$ is differentiable. Explain the practical significance of your answers.

(d) Show algebraically that $S\left(t\right)$ is a continuous function, but not a differentiable function.

Problem Zero: Read the section and make your own sum-

mary of the material.

True/False: Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample.

Part (a): The slope of the tangent line to a function f at the point $x = 4$ is given by $f\text{'}\left(4\right)$.

Part (b): The instantaneous rate of change of a function f at the point $x=-3$ is given by $f(-3)$.

Part (c): The instantaneous rate of change of a function f at a point $x=a$ can be represented as the slope of a secant line.

Part (d): Where a function f is positive, its associated slope function f ' is increasing.

Part (e): Where a function f is decreasing, its associated slope function f ' is negative.

Part (f): When a function f has a steep slope at a point on its graph, its instantaneous rate of change at that point will have a large magnitude.

Part (g): When the graph of a function f  is decreasing with a steep slope, the graph of the associated slope function f  is negative with a large magnitude.

Part (h): Suppose an object is moving in a straight path with position function $s\left(t\right)$. If data-custom-editor="chemistry" $\mathrm{s}\left(\mathrm{t}\right)$ is positive and decreasing, then the velocity $v\left(t\right)$ is negative.

Slope and linear functions: If f is a linear function with slope $-3$such that $f\left(2\right)=1,$find the following, without first finding an equation for $f\left(x\right).$

$$\begin{gathered} f\left(4\right) \\ f\left(7\right) \\ f(-2) \end{gathered}$$

Examples: Construct examples of the thing(s) described in the following. Try to find examples that are different than any in the reading. 

(a) The graph of a function whose associated slope function $f^{\text{'}}$ is positive on (−∞, 2) and negative on (2,∞).

(b) The graph of a function with the following three properties: The average rate of change of f on [0, 2] is 3, the average rate of change of f on [0, 1] is −1, and the average rate of change of f on [−2, 2] is 0. 

(c) The graph of a function f with the following three properties: The instantaneous rate of change of f at x = 2 is zero, the average rate of change of f on [1, 2] is 2, and the average rate of change of f on [2, 4] is 1. 

Approximating limits: Use sequences of approximations to estimate the values of

$$\begin{gathered} \underset{x\to 2}{\lim }\frac{4-x^2}{2-x} \\ \underset{z\to 3}{\lim }\frac{z^3-27}{z-3} \end{gathered}$$

Explain why it is not a simple task to calculate the slope of the tangent line to a function $f$ at a point $x=c$. Shouldn’t calculating the slope of a line be really easy? What goes wrong here?

Identifying increasing and decreasing behavior: Use a graphing utility to determine the intervals on which $f\left(x\right)=-4x^5+25x^4-40x^3$is increasing or decreasing.

When flying home for the holidays, Eva often flies between Denver International Airport (DLA) and Chicago O'Hare (ORD). Suppose Eva's plane takes off from DIA and 50 miles from ORD the plane has to circle the airport because of snow. The plane circles ORD four times and then lands.

(a) Draw a graph depicting the distance from DIA to Eva's plane as a function of time.

(b) Draw a graph depicting the distance from ORD to Eva's plane as a function of time.

Let $l$ be the line connecting two points $(a,f(a\left)\right)$ and $(b,f(b\left)\right)$ on the graph of a function  $f$. What does this line $l$ have to do with the average rate of change of $f$ on the interval $[a,b],$ and why?

Given that $s\left(t\right)$ measures the distance an object has travelled over time, explain what the expression $\frac{s\left(b\right)-s\left(a\right)}{b-a}$ has to do with the distance formula $d=rt$.

How is velocity different from speed? What does it mean if velocity is negative?

What is the relationship between the derivative of a function $f$ at a point $x=c$, the slope of the tangent line to the graph of $f$ at $x=c$, and the instantaneous rate of change of $f$ at $x=c$?

On a graph of f(x) = $x^2$,

(a) draw the tangent line to the graph of f at the point (2, f(2)); 

(b) draw the secant line from (2, f(2)) to (2.75, f(2.75)); 

(c) draw the secant line from (1.75, f(1.75)) to (2, f(2)). 

(d) Which secant line is a better approximation to the tangent line, and why? 

In Example 3 we estimated the slope of the tangent line to $f\left(x\right)=-\frac{1}{2}{x}^2+3x$ at $x=2$. Get a better estimate by calculating the slopes of secant lines with values of $z$ even closer to $x=2$, for example, $z=2.01,z=2.001$, and $z=2.0001$.

In Example 3 we estimated the slope of the tangent line to $f\left(x\right)=-\frac{1}{2}{x}^2+3x\text{ at }x=2$ by finding slopes of secant lines from x = 2 to various points x = z with z > 2. Draw a sequence of graphs that illustrates how to do this for z < 2, and then make specific calculations for z = 1, z = 1.5, z = 1.75, and z = 1.9. What are the corresponding values of h in this example? 

For the graph of f appearing next at the left, label each of the following quantities to illustrate that 

$f^{\mathrm{\prime }}\left(c\right)\approx \frac{f(c+h)-f\left(c\right)}{h}$

(a) the locations c, c + h, f(c), and f(c + h) 

(b) the distances h and f(c + h) − f(c) 

(c) the slopes $\frac{f(c+h)-f\left(c\right)}{h}$ and $f^{\mathrm{\prime }}\left(c\right)$

For the graph of g(x) appearing next at the right, label each of the following quantities to illustrate that 

$g^{\mathrm{\prime }}\left(c\right)\approx \frac{g(c+h)-g\left(c\right)}{h}$

(a) the locations c, c + h, g(c), and g(c + h) 

(b) the distances h and g(c + h) − g(c) 

(c) the slopes $\frac{g(c+h)-g\left(c\right)}{h}$ and $g^{\mathrm{\prime }}\left(c\right)$

Consider again the graph of f at the left. Label each of the following quantities to illustrate that 

$f^{\mathrm{\prime }}\left(c\right)\approx \frac{f\left(z\right)-f\left(c\right)}{z-c}$

(a) the locations c, z, f(c), and f(z) 

(b) the distances z − c and f(z) − f(c) 

(c) the slopes $\frac{f\left(z\right)-f\left(c\right)}{z-c}$ and $f^{\mathrm{\prime }}\left(c\right)$

Consider again the graph of g(x) at the right. Label each of the following quantities to illustrate that 

$g^{\text{'}}\left(c\right)\approx \frac{g\left(z\right)-g\left(c\right)}{z-c}$

(a) the locations c, z, g(c), and g(z) 

(b) the distances z − c and g(z) − g(c) 

(c) the slopes $\frac{g\left(z\right)-g\left(c\right)}{z-c}$ and $g^{\text{'}}\left(c\right)$

For the graph of f shown next at the left, list the following quantities in order from least to greatest:

(a) the average rate of change of f on $[-1,1]$

(b) the instantaneous rate of change of f at $x=1$

(c) $f\text{'}(-1)$

(d) $\frac{f\left(2\right)-f(-1)}{2-(-1)}$

For the graph of g(x) shown next at the right, list the following quantities in order from least to greatest:
(a) the average rate of change of g on $[0,1]$
(b) the instantaneous rate of change of g at $x=1$ is $0.4$
(c) $\frac{g(-1+0.1)-g(-1)}{0.1}$

(d) $\frac{g\left(1\right)-g(-1)}{1-(-1)}=$

Consider again the function f graphed at the left. At which values of x does f have the greatest instantaneous rate of change? The least? At which values of x is the
instantaneous rate of change of f equal to zero?

Consider again the function g(x) graphed at the right. For which values of x does g(x) have a positive instantaneous rate of change? Negative? Zero?

Make a copy of the graph of f used in Exercises 11 and 13, and sketch additional secant lines to illustrate that as $h\to 0$ (or equivalently, as $z\to c$) the slopes of the secant line get closer and closer to the slope of the tangent line to f at $x=c$.

The derivative of a smooth function f at a point $x=c$ can also be approximated with a symmetric difference quotient:

$f\text{'}\left(c\right)\approx \frac{f(c+h)-f(c-h)}{2h}$

(a) Use a graph to illustrate what the symmetric difference measures. Why would it be reasonable to use the two-sided symmetric difference to approximate f'(c)? (Hint: Your answer should involve a certain kind of secant line and a discussion of what happens as h gets close to $0$.)

(b) Use a sequence of symmetric difference approximations to estimate the derivative of $f\left(x\right)=x^2 at x=3$ Illustrate your answer with a sequence of graphs.

In Exercises 21–24, sketch the graph of a function f that has the listed characteristics. 

$$\begin{gathered} f\text{'}(-3)=0; \\ f\text{'}(-1)=0; \\ f\text{'}\left(2\right)=0 \end{gathered}$$

In Exercises 21–24, sketch the graph of a function f that has the listed characteristics.
$$\begin{gathered} f\text{'}(-2)=2; \\ f\text{'}\left(0\right)=1; \\ f\left(1\right)=-5 \end{gathered}$$

In Exercises 21–24, sketch the graph of a function f that has the listed characteristics.

$$\begin{gathered} f\left(1\right)=2; \\ f\text{'}\left(1\right)=0; \\ f\text{'}\left(3\right)=2 \end{gathered}$$

In Exercises 21–24, sketch the graph of a function f that has the listed characteristics.
$$\begin{gathered} f(-1)=2; \\ f\text{'}(-1)=3; \\ f\left(1\right)=-2; \\ f\text{'}\left(1\right)=3 \end{gathered}$$

Sketch a graph of the associated slope function f for each function f in Exercises 25–30.

Sketch a graph of the associated slope function f for each function f in Exercises 25–30.

Sketch a graph of the associated slope function f for each function f in Exercises 25–30.

Sketch a graph of the associated slope function f for each function f in Exercises 25–30.

Sketch a graph of the associated slope function f for each function f in Exercises 25–30.

Sketch a graph of the associated slope function f' for each function f.

Each graph in Exercises 31–34 can be thought of as the associated slope function f' for some unknown function f. In each case sketch a possible graph of f.

Each graph in Exercises 31–34 can be thought of as the associated slope function f' for some unknown function f. In each case sketch a possible graph of f. 

Each graph in Exercises 31–34 can be thought of as the associated slope function f' for some unknown function f. In each case sketch a possible graph of f.  

Each graph in Exercises 31–34 can be thought of as the associated slope function f' for some unknown function f. In each case sketch a possible graph of f.  

For each function f and value $x=c$ in Exercises 35–44, use a sequence of approximations to estimate $f\text{'}\left(c\right)$. Illustrate your work with an appropriate sequence of graphs of secant lines.

$f\left(x\right)=4-x^2, c=1$

For each function f and value $x=c$ in Exercises 35–44, use a sequence of approximations to estimate $f\text{'}\left(c\right)$. Illustrate your work with an appropriate sequence of graphs of secant lines.

$f\left(x\right)=4-x^2, c=0$

For each function f and value $x=c$ in Exercises 35–44, use a sequence of approximations to estimate $f\text{'}\left(c\right)$. Illustrate your work with an appropriate sequence of graphs of secant lines.

$f\left(x\right)=x+x^3, c=0$ 

For each function f and value $x=c$ in Exercises 35–44, use a sequence of approximations to estimate $f\text{'}\left(c\right)$. Illustrate your work with an appropriate sequence of graphs of secant lines.

$f\left(x\right)=x+x^3, c=1$

For each function f and value $x=c$ in Exercises 35–44, use a sequence of approximations to estimate $f\text{'}\left(c\right)$. Illustrate your work with an appropriate sequence of graphs of secant lines.

$f\left(x\right)=\ln \left(x^2+1\right), c=0$

For each function f and value x = c in Exercises 35–44, use a sequence of approximations to estimate $f^{\text{'}}\left(c\right)$. Illustrate your work with an appropriate sequence of graphs of secant lines.

$f\left(x\right)=\sin x,c=\frac{\pi }{2}$

For each function f and value x = c in Exercises 35–44, use a sequence of approximations to estimate $f^{\text{'}}\left(c\right)$. Illustrate your work with an appropriate sequence of graphs of secant lines.

$f\left(x\right)=\arctan x,c=0$