12.4 Derivatives

Find the average rate of change connecting the points $\left(2 , −6\right)$ and $\left(−1 , 5\right) .$

Find the average rate of change connecting the points $\left(- 5 , 1.5\right)$ and $\left(- 2.5 , 9\right) .$

Find the derivative of the function $f \left(x\right) = x^{2} - 3 x + 5$ at $x = a .$

Find the derivative of the function $f \left(x\right) = 3 x^{2} + 7 x$ at $x = a .$

Find the derivative of the function $f \left(x\right) = \frac{3 + x}{2 - x}$ at $x = a .$

Find the derivative of the function $f \left(x\right) = \frac{10 x + 11}{5 x + 4}$ at $x = a .$

Find the derivative of the function $f \left(x\right) = 4 \sqrt{x}$ at $x = 36.$

Find the derivative of the function $f \left(x\right) = 9 \sqrt{x}$ at $x = 9.$

Using the function above, $s \left(t\right) = −16 t^{2} + 64 t + 6 ,$ what is the instantaneous velocity of the ball at 1 second and 3 seconds into its flight?

The position of the ball is given by $s \left(t\right) = −16 t^{2} + 64 t + 6.$ What is its velocity 2 seconds into flight?

The cost in dollars of producing $x$ laptop computers in dollars is $f \left(x\right) = x^{2} - 100 x .$ At the point where 200 computers have been produced, what is the approximate cost of producing the 201^st unit?

A car leaves an intersection. The distance it travels in miles is given by the function $f \left(t\right) ,$ where $t$ represents hours. Explain the following notations:

1. ⓐ $f \left(0\right) = 0$
2. ⓑ $f^{'} \left(1\right) = 60$
3. ⓒ $f \left(1\right) = 70$
4. ⓓ $f \left(2.5\right) = 150$

A runner runs along a straight east-west road. The function $f \left(t\right)$ gives how many feet eastward of her starting point she is after $t$ seconds. Interpret each of the following as it relates to the runner.

1. ⓐ $f \left(0\right) = 0$
2. ⓑ $f \left(10\right) = 150$
3. ⓒ $f^{'} \left(10\right) = 15$
4. ⓓ $f^{'} \left(20\right) = - 10$
5. ⓔ $f \left(40\right) = −100$

Find the equation of a line tangent to the curve $f \left(x\right) = x^{2} - 4 x$ at $x = 3.$

Find the equation of a tangent line to the curve of the function $f \left(x\right) = 5 x^{2} - x + 4$ at $x = 2.$

A ball is tossed upward from a height of 200 feet with an initial velocity of 36 ft/sec. If the height of the ball in feet after $t$ seconds is given by $s \left(t\right) = −16 t^{2} + 36 t + 200 ,$ find the instantaneous velocity of the ball at $t = 2.$

A fireworks rocket is shot upward out of a pit 12 ft below the ground at a velocity of 60 ft/sec. Its height in feet after $t$ seconds is given by $s = - 16 t^{2} + 60 t - 12.$ What is its instantaneous velocity after 4 seconds?

How is the slope of a linear function similar to the derivative?

A car traveled 110 miles during the time period from 2:00 P.M. to 4:00 P.M. What was the car's average velocity? At exactly 2:30 P.M., the speed of the car registered exactly 62 miles per hour. What is another name for the speed of the car at 2:30 P.M.? Why does this speed differ from the average velocity?

Suppose water is flowing into a tank at an average rate of 45 gallons per minute. Translate this statement into the language of mathematics.

Use the definition of derivative $\underset{h \rightarrow 0}{\lim} \frac{f \left(x + h\right) - f \left(x\right)}{h}$ to calculate the derivative of each function.

$f \left(x\right) = - 2 x + 1$

Use the definition of derivative $\underset{h \rightarrow 0}{\lim} \frac{f \left(x + h\right) - f \left(x\right)}{h}$ to calculate the derivative of each function.

$f \left(x\right) = 2 x^{2} + x - 3$

Use the definition of derivative $\underset{h \rightarrow 0}{\lim} \frac{f \left(x + h\right) - f \left(x\right)}{h}$ to calculate the derivative of each function.

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

Use the definition of derivative $\underset{h \rightarrow 0}{\lim} \frac{f \left(x + h\right) - f \left(x\right)}{h}$ to calculate the derivative of each function.

$f \left(x\right) = \frac{5 - 2 x}{3 + 2 x}$

Use the definition of derivative $\underset{h \rightarrow 0}{\lim} \frac{f \left(x + h\right) - f \left(x\right)}{h}$ to calculate the derivative of each function.

$f \left(x\right) = 3 x^{3} - x^{2} + 2 x + 5$

Use the definition of derivative $\underset{h \rightarrow 0}{\lim} \frac{f \left(x + h\right) - f \left(x\right)}{h}$ to calculate the derivative of each function.

$f \left(x\right) = 5 \pi$

Find the average rate of change between the two points.

$\left(4 , −3\right)$ and $\left(−2 , −1\right)$

Find the average rate of change between the two points.

$\left(7 , −2\right)$ and $\left(7 , 10\right)$

For the following polynomial functions, find the derivatives.

$f \left(x\right) = - 3 x^{2} - 7 x = 6$

For the following polynomial functions, find the derivatives.

$f \left(x\right) = 3 x^{3} + 2 x^{2} + x - 26$

For the following functions, find the equation of the tangent line to the curve at the given point $x$ on the curve.

$\begin{matrix}f \left(x\right) = x^{3} + 1 & x = 2\end{matrix}$

Find $k$ such that the given line is tangent to the graph of the function.

$\begin{matrix}f \left(x\right) = x^{2} - k x , & y = 4 x - 9\end{matrix}$

$f \left(0\right)$

$f \left(2\right)$

$f^{'} \left(- 1\right)$

$f^{'} \left(1\right)$

$f^{'} \left(3\right)$

Numerically evaluate the derivative. Explore the behavior of the graph of $f \left(x\right) = x^{2}$ around $x = 1$ by graphing the function on the following domains: $\left[0.9 , 1.1\right]$, $\left[0.99 , 1.01\right]$, $\left[0.999 , 1.001\right] ,$ and $\left[0.9999 , 1.0001\right]$ . We can use the feature on our calculator that automatically sets Ymin and Ymax to the Xmin and Xmax values we preset. (On some of the commonly used graphing calculators, this feature may be called ZOOM FIT or ZOOM AUTO). By examining the correspond…

Explain the notation in words. The volume $f \left(t\right)$ of a tank of gasoline, in gallons, $t$ minutes after noon.

$f ' \left(30\right) = −20$

Explain the notation in words. The volume $f \left(t\right)$ of a tank of gasoline, in gallons, $t$ minutes after noon.

$f ' \left(200\right) = 30$

Explain the functions in words. The height, $s ,$ of a projectile after $t$ seconds is given by $s \left(t\right) = - 16 t^{2} + 80 t .$

$s \left(2\right) = 96$

Explain the functions in words. The height, $s ,$ of a projectile after $t$ seconds is given by $s \left(t\right) = - 16 t^{2} + 80 t .$

$s \left(3\right) = 96$

Explain the functions in words. The height, $s ,$ of a projectile after $t$ seconds is given by $s \left(t\right) = - 16 t^{2} + 80 t .$

$s \left(0\right) = 0 , s \left(5\right) = 0.$

The volume $V$ of a sphere with respect to its radius $r$ is given by $V = \frac{4}{3} \pi r^{3} .$

Find the instantaneous rate of change of $V$ when $r = 3 \textrm{ }\text{cm} .$

The revenue generated by selling $x$ items is given by $R \left(x\right) = 2 x^{2} + 10 x .$

Find $R ' \left(10\right)$ and interpret.

The cost of producing $x$ cellphones is described by the function $C \left(x\right) = x^{2} - 4 x + 1000.$

Find the average rate of change in the total cost as $x$ changes from $x = 10 \textrm{ }\text{to}\textrm{ } x = 15.$

The cost of producing $x$ cellphones is described by the function $C \left(x\right) = x^{2} - 4 x + 1000.$

Find the approximate marginal cost, when 20 cellphones have been produced, of producing the 21^st cellphone.

Use the definition for the derivative at a point $x = a ,$ $\underset{x \rightarrow a}{\lim} \frac{f \left(x\right) - f \left(a\right)}{x - a} ,$ to find the derivative of the functions.

$f \left(x\right) = 5 x^{2} - x + 4$

Use the definition for the derivative at a point $x = a ,$ $\underset{x \rightarrow a}{\lim} \frac{f \left(x\right) - f \left(a\right)}{x - a} ,$ to find the derivative of the functions.

$f \left(x\right) = \frac{- 4}{3 - x^{2}}$

$\underset{x \rightarrow −1^{+}}{\lim} f \left(x\right)$

$\underset{x \rightarrow - 1}{\lim} f \left(x\right)$

At what values of $x$ is the function discontinuous? What condition of continuity is violated?

With the use of a graphing utility, use numerical or graphical evidence to determine the left- and right-hand limits of the function given as $x$ approaches $a .$ If the function has limit as $x$ approaches $a ,$ state it. If not, discuss why there is no limit.

$f \left(x\right) = \left\{\right. \begin{matrix}\left|x\right| - 1 , & i f & x \neq 1 \\ x^{3} , & i f & x = 1\end{matrix} \textrm{ }\textrm{ } a = 1$

With the use of a graphing utility, use numerical or graphical evidence to determine the left- and right-hand limits of the function given as $x$ approaches $a .$ If the function has limit as $x$ approaches $a ,$ state it. If not, discuss why there is no limit.

$f \left(x\right) = \left\{\right. \begin{matrix}\sqrt{x + 3} , & i f & x < 1 \\ - \sqrt[3]{x} , & i f & x > 1\end{matrix} \textrm{ }\textrm{ } a = 1$

Find the limits if $\underset{x \rightarrow c}{\lim} f \left(x\right) = −3$ and $\underset{x \rightarrow c}{\lim} g \left(x\right) = 5.$

$\underset{x \rightarrow c}{\lim} \frac{f \left(x\right)}{g \left(x\right)}$

Find the limits if $\underset{x \rightarrow c}{\lim} f \left(x\right) = −3$ and $\underset{x \rightarrow c}{\lim} g \left(x\right) = 5.$

$\underset{x \rightarrow 0^{+}}{\lim} f \left(x\right) , f \left(x\right) = \left\{\right. \begin{matrix}3 x^{2} + 2 x + 1 \\ 5 x + 3\end{matrix} \textrm{ }\textrm{ } \begin{matrix}x > 0 \\ x < 0\end{matrix}$

Find the limits if $\underset{x \rightarrow c}{\lim} f \left(x\right) = −3$ and $\underset{x \rightarrow c}{\lim} g \left(x\right) = 5.$

$\underset{x \rightarrow 3^{+}}{\lim} \left(3 x - \left[x\right]\right)$

Evaluate the limits using algebraic techniques.

$\underset{x \rightarrow 25}{\lim} \left(\frac{x^{2} - 625}{\sqrt{x} - 5}\right)$

Evaluate the limits using algebraic techniques.

$\begin{matrix}\lim \\ ^{x \rightarrow 4}\end{matrix} \frac{7 - \sqrt{12 x + 1}}{x - 4}$

Use numerical evidence to determine whether the limit exists at $x = a .$ If not, describe the behavior of the graph of the function at $x = a .$

$f \left(x\right) = \frac{- 2}{x - 4} ; a = 4$

Use numerical evidence to determine whether the limit exists at $x = a .$ If not, describe the behavior of the graph of the function at $x = a .$

$f \left(x\right) = \frac{- x}{x^{2} - x - 6} ; a = 3$

Use numerical evidence to determine whether the limit exists at $x = a .$ If not, describe the behavior of the graph of the function at $x = a .$

$f \left(x\right) = \frac{\sqrt{x} - 3}{9 - x} ; a = 9$

Determine where the given function $f \left(x\right)$ is continuous. Where it is not continuous, state which conditions fail, and classify any discontinuities.

$f \left(x\right) = \frac{x^{2} - 2 x - 15}{x - 5}$

Determine where the given function $f \left(x\right)$ is continuous. Where it is not continuous, state which conditions fail, and classify any discontinuities.

$f \left(x\right) = \frac{x^{3} - 125}{2 x^{2} - 12 x + 10}$

Determine where the given function $f \left(x\right)$ is continuous. Where it is not continuous, state which conditions fail, and classify any discontinuities.

$f \left(x\right) = \frac{x + 2}{x^{2} - 3 x - 10}$

Find the average rate of change $\frac{f \left(x + h\right) - f \left(x\right)}{h} .$

$f \left(x\right) = 3 x + 2$

Find the average rate of change $\frac{f \left(x + h\right) - f \left(x\right)}{h} .$

$f \left(x\right) = \frac{1}{x + 1}$

Find the average rate of change $\frac{f \left(x + h\right) - f \left(x\right)}{h} .$

$f \left(x\right) = e^{2 x}$

Find the derivative of the function.

$f \left(x\right) = 5 x^{2} - 3 x$

With the aid of a graphing utility, explain why the function is not differentiable everywhere on its domain. Specify the points where the function is not differentiable.

$f \left(x\right) = \frac{x}{\left|x\right|}$

$f \left(1\right)$

$\underset{x \rightarrow −1^{-}}{\lim} f \left(x\right)$

$\underset{x \rightarrow −2}{\lim} f \left(x\right)$

With the use of a graphing utility, use numerical or graphical evidence to determine the left- and right-hand limits of the function given as $x$ approaches $a .$ If the function has a limit as $x$ approaches $a ,$ state it. If not, discuss why there is no limit

$f \left(x\right) = \left\{\right. \begin{matrix}\frac{1}{x} - 3 , \textrm{ }\text{i} f & x \leq 2 \\ x^{3} + 1 , i f & x > 2\end{matrix} \textrm{ }\textrm{ } a = 2$

Evaluate each limit using algebraic techniques.

$\underset{h \rightarrow 0}{\lim} \left(\frac{\sqrt{h^{2} + 25} - 5}{h^{2}}\right)$

Determine whether or not the given function $f$ is continuous. If it is continuous, show why. If it is not continuous, state which conditions fail.

$f \left(x\right) = \sqrt{x^{2} - 4}$

Use the definition of a derivative to find the derivative of the given function at $x = a .$

$f \left(x\right) = \frac{3}{5 + 2 x}$

Use the definition of a derivative to find the derivative of the given function at $x = a .$

$f \left(x\right) = 2 x^{2} + 9 x$

With the aid of a graphing utility, explain why the function is not differentiable everywhere on its domain. Specify the points where the function is not differentiable.

$f \left(x\right) = \left|x - 2\right| - \left|x + 2\right|$

Explain the notation in words when the height of a projectile in feet, $s ,$ is a function of time $t$ in seconds after launch and is given by the function $s \left(t\right) .$

$s \left(0\right)$

Explain the notation in words when the height of a projectile in feet, $s ,$ is a function of time $t$ in seconds after launch and is given by the function $s \left(t\right) .$

$s ' \left(2\right)$

Explain the notation in words when the height of a projectile in feet, $s ,$ is a function of time $t$ in seconds after launch and is given by the function $s \left(t\right) .$

$s \left(t\right) = 0$

Use technology to evaluate the limit.

$\underset{x \rightarrow 0}{\lim} \frac{\tan^{2} \left(x\right)}{2 x}$

Use technology to evaluate the limit.

Evaluate the limit by hand.

$\begin{matrix}\lim \\ ^{x \rightarrow 1}\end{matrix} f \left(x\right) , \textrm{ }\text{where}\textrm{ }\textrm{ } f \left(x\right) = \left\{\right. \begin{matrix}4 x - 7 & x \neq 1 \\ x^{2} - 4 & x = 1\end{matrix}$

At what value(s) of $x$ is the function below discontinuous?

$f \left(x\right) = \left\{\right. \begin{matrix}4 x - 7 x \neq 1 \\ x^{2} - 4 x = 1\end{matrix}$

Find the average rate of change of the function from $x = 1 \textrm{ }\text{to}\textrm{ } x = 3.$

Find all values of $x$ at which $f ' \left(x\right) = 0.$

Find an equation of the tangent line to the graph of $f$ the indicated point: $f \left(x\right) = 3 x^{2} - 2 x - 6 , \textrm{ }\textrm{ } x = - 2$

Use the function $f \left(x\right) = x \left(1 - x\right)^{\frac{2}{5}}$.

Explore the behavior of the graph of $f \left(x\right)$ around $x = 1$ by graphing the function on the following domains, [0.9, 1.1], [0.99, 1.01], [0.999, 1.001], and [0.9999, 1.0001]. Use this information to determine whether the function appears to be differentiable at $x = 1.$

Find the derivative of each of the functions using the definition: $\underset{h \rightarrow 0}{\lim} \frac{f \left(x + h\right) - f \left(x\right)}{h}$

$f \left(x\right) = 4 x^{2} - 7$

Find the derivative of each of the functions using the definition: $\underset{h \rightarrow 0}{\lim} \frac{f \left(x + h\right) - f \left(x\right)}{h}$

$f \left(x\right) = \frac{1}{x + 2}$

Find the derivative of each of the functions using the definition: $\underset{h \rightarrow 0}{\lim} \frac{f \left(x + h\right) - f \left(x\right)}{h}$

$f \left(x\right) = - x^{3} + 1$

Find the derivative of each of the functions using the definition: $\underset{h \rightarrow 0}{\lim} \frac{f \left(x + h\right) - f \left(x\right)}{h}$

$f \left(x\right) = \sqrt{x - 1}$