3.1 Defining the derivative  (Page 6/10)

$f\left(x\right)=4x+7;x_1=2,x_2=5$

$f\left(x\right)=8x-3;x_1=\mathrm{-1},x_2=3$

$f\left(x\right)=x^2+2x+1;x_1=3,x_2=3.5$

$f\left(x\right)=\text{−}{x}^2+x+2;x_1=0.5,x_2=1.5$

$f\left(x\right)=\frac{4}{3x-1};x_1=1,x_2=3$

$f\left(x\right)=\frac{x-7}{2x+1};x_1=\mathrm{-2},x_2=0$

$f\left(x\right)=\sqrt{x};x_1=1,x_2=16$

$f\left(x\right)=\sqrt{x-9};x_1=10,x_2=13$

$f\left(x\right)=x^{1\text{/}3}+1;x_1=0,x_2=8$

$f\left(x\right)=6x^{2\text{/}3}+2x^{1\text{/}3};x_1=1,x_2=27$

$f\left(x\right)=3-4x,a=2$

$f\left(x\right)=\frac{x}{5}+6,a=\mathrm{-1}$

$f\left(x\right)=x^2+x,a=1$

$f\left(x\right)=1-x-x^2,a=0$

$f\left(x\right)=\frac{7}{x},a=3$

$f\left(x\right)=\sqrt{x+8},a=1$

$f\left(x\right)=2-3x^2,a=\mathrm{-2}$

$f\left(x\right)=\frac{\mathrm{-3}}{x-1},a=4$

$f\left(x\right)=\frac{2}{x+3},a=\mathrm{-4}$

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

$f\left(x\right)=5x+4,a=\mathrm{-1}$

$f\left(x\right)=\mathrm{-7}x+1,a=3$

$f\left(x\right)=x^2+9x,a=2$

$f\left(x\right)=3x^2-x+2,a=1$

$f\left(x\right)=\sqrt{x},a=4$

$f\left(x\right)=\sqrt{x-2},a=6$

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

$f\left(x\right)=\frac{1}{x-3},a=\mathrm{-1}$

$f\left(x\right)=\frac{1}{x^3},a=1$

$f\left(x\right)=\frac{1}{\sqrt{x}},a=4$

[T] $f\left(x\right)=x^2+3x+4,P\left(1,8\right)$ (Round to $6$ decimal places.)

[T] $f\left(x\right)=\frac{x+1}{x^2-1},P\left(0,\mathrm{-1}\right)$

[T] $f\left(x\right)=10e^{0.5x},P\left(0,10\right)$ (Round to $4$ decimal places.)

[T] $f\left(x\right)=\text{tan}\left(x\right),P\left(\pi ,0\right)$

$s\left(t\right)=\frac{1}{3}t+5$

$s\left(t\right)=t^2-2t$

$s\left(t\right)=2t^3+3$

$s\left(t\right)=\frac{16}{t^2}-\frac{4}{t}$

Use the following graph to evaluate a. $f^{\prime }\left(1\right)$ and b. $f^{\prime }\left(6\right).$

Use the following graph to evaluate a. $f^{\prime }\left(\mathrm{-3}\right)$ and b. $f^{\prime }\left(1.5\right).$

$f\left(x\right)=x^{1\text{/}3},x=0$

$f\left(x\right)=x^{2\text{/}3},x=0$

$f\left(x\right)=\{\begin{array}{c}1,x<1\\ x,x\ge 1\end{array},x=1$

$f\left(x\right)=\frac{\left|x\right|}{x},x=0$

[T] The position in feet of a race car along a straight track after $t$ seconds is modeled by the function $s\left(t\right)=8t^2-\frac{1}{16}{t}^3.$

1. Find the average velocity of the vehicle over the following time intervals to four decimal places:
   1. [4, 4.1]
   2. [4, 4.01]
   3. [4, 4.001]
   4. [4, 4.0001]
2. Use a. to draw a conclusion about the instantaneous velocity of the vehicle at $t=4$ seconds.

[T] The distance in feet that a ball rolls down an incline is modeled by the function $s\left(t\right)=14t^2,$ where t is seconds after the ball begins rolling.

1. Find the average velocity of the ball over the following time intervals:
   1. [5, 5.1]
   2. [5, 5.01]
   3. [5, 5.001]
   4. [5, 5.0001]
2. Use the answers from a. to draw a conclusion about the instantaneous velocity of the ball at $t=5$ seconds.

Two vehicles start out traveling side by side along a straight road. Their position functions, shown in the following graph, are given by $s=f(t)$ and $s=g\left(t\right),$ where $s$ is measured in feet and $t$ is measured in seconds.

1. Which vehicle has traveled farther at $t=2$ seconds?
2. What is the approximate velocity of each vehicle at $t=3$ seconds?
3. Which vehicle is traveling faster at $t=4$ seconds?
4. What is true about the positions of the vehicles at $t=4$ seconds?

[T] The total cost $C(x),$ in hundreds of dollars, to produce $x$ jars of mayonnaise is given by $C\left(x\right)=0.000003x^3+4x+300.$

1. Calculate the average cost per jar over the following intervals:
   1. [100, 100.1]
   2. [100, 100.01]
   3. [100, 100.001]
   4. [100, 100.0001]
2. Use the answers from a. to estimate the average cost to produce $100$ jars of mayonnaise.

[T] For the function $f\left(x\right)=x^3-2x^2-11x+12,$ do the following.

1. Use a graphing calculator to graph f in an appropriate viewing window.
2. Use the ZOOM feature on the calculator to approximate the two values of $x=a$ for which $m_{\text{tan}}=f^{\prime }\left(a\right)=0.$

[T] For the function $f\left(x\right)=\frac{x}{1+x^2},$ do the following.

1. Use a graphing calculator to graph $f$ in an appropriate viewing window.
2. Use the ZOOM feature on the calculator to approximate the values of $x=a$ for which $m_{\text{tan}}=f^{\prime }\left(a\right)=0.$

Suppose that $N\left(x\right)$ computes the number of gallons of gas used by a vehicle traveling $x$ miles. Suppose the vehicle gets $30$ mpg.

1. Find a mathematical expression for $N\left(x\right).$
2. What is $N(100\text{)?}$ Explain the physical meaning.
3. What is $N^{\prime }(100)?$ Explain the physical meaning.

[T] For the function $f\left(x\right)=x^4-5x^2+4,$ do the following.

1. Use a graphing calculator to graph $f$ in an appropriate viewing window.
2. Use the $\text{nDeriv}$ function, which numerically finds the derivative, on a graphing calculator to estimate $f^{\prime }\left(\mathrm{-2}\right),f^{\prime }(\mathrm{-0.5}),f^{\prime }(1.7),$ and $f^{\prime }(2.718).$

[T] For the function $f\left(x\right)=\frac{x^2}{x^2+1},$ do the following.

1. Use a graphing calculator to graph $f$ in an appropriate viewing window.
2. Use the $\text{nDeriv}$ function on a graphing calculator to find $f^{\prime }\left(\mathrm{-4}\right),f^{\prime }(\mathrm{-2}),f^{\prime }(2),$ and $f^{\prime }(4).$