3.1: Defining the Derivative

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A car travels down a straight road in the direction of Phoenix, Arizona. At 12:38 pm, the car passes a sign that says Phoenix is 51 miles away. At 1:08 pm, the car passes a sign that says Phoenix is 15 miles away. Calculate the average velocity of the car between 12:38 pm and 1:08 pm.

Caleb throws a ball into the air, then the ball falls back to the ground. The height of the ball above the ground, in meters, $t$ seconds after Caleb releases it is given by the function $y(t) = -4.9t^2 + 10.3t + 1.8$. Use a calculator to calculate the average velocity of the ball over each time interval in the table below, then use your results to estimate the instantaneous velocity of the ball at $t = 2$ seconds.

Interval	Average Velocity		Interval	Average Velocity
[1.9, 2]			[2, 2.1]
[1.99, 2]			[2, 2.01]
[1.999, 2]			[2, 2.001]

Crystal can buy a package of 12 chip bags for $8.00 or a package of 18 chip bags for $15.00. What is the average cost per bag Crystal would pay for the additional 6 chip bags that come in the larger package?

The ABC Demolition Company estimates that the total cost for them to dispose of $x$ pounds of rubble during a small demolition project is given by the function $C(x) = 2\sqrt{x} + 150$. Use a calculator to calculate the average disposal cost per pound of rubble over each interval in the table below, then use your results to estimate the disposal cost per pound of rubble when the company has disposed of exactly $x = 400$ pounds of rubble.

Interval	Average Cost per Pound		Interval	Average Cost per Pound
[399, 400]			[400, 401]
[399.9, 400]			[400, 400.1]
[399.99, 400]			[400, 400.01]

If $f(2) = 10$ and $f(5) = 3$ for some function $f(x)$, find the average rate of change of the function over the interval $[2, 5]$.
If $g(-3) = 2$ and $g(1) = 7$ for some function $g(x)$, find the slope of the secant line passing through the function at $x = -3$ and $x = 1$.

Let $h(x) = \dfrac{1}{2}x^2$. Use a calculator to calculate the average rate of change of the function over each interval in the table below, then use your results to estimate the slope of the tangent line to the function at $x = 3$.

Interval	Average Rate of Change		Interval	Average Rate of Change
[2.9, 3]			[3, 3.1]
[2.99, 3]			[3, 3.01]
[2.999, 3]			[3, 3.001]

Using a graphing calculator, plot the graph of $h(x) = -2x + 3$, then estimate the instantaneous rate of change of the function at $x = 2$.
Using a graphing calculator, plot the graph of $f(x) = \sqrt{x}$, then estimate the slope of the tangent line at $x = 1$.
Using a graphing calculator, plot the graph of $g(x) = -x^2 + 2x$, then estimate the instantaneous rate of change of the function at $x = 1$.
Using a graphing calculator, plot the graph of $h(x) = \sin x$, then estimate the slope of the tangent line at $x = 0$.
Let $f(x) = 5x + 2$. Calculate the exact value of the slope of the tangent line at $x = -1$.
Let $g(x) = 2x^2 - 3$. Calculate the exact value of the slope of the tangent line at $x = 2$.
Let $h(x) = \sqrt{x + 1}$. Calculate the exact value of the slope of the tangent line at $x = 3$.
Let $f(x) = \dfrac{2}{x}$. Calculate the exact value of the slope of the tangent line at $x = 1$.
Let $g(x) = x^2 + x$. Find the equation of the line tangent to $g(x)$ at $x = 0$.
Let $h(x) = \dfrac{1}{\sqrt{x}}$. Find the equation of the line tangent to $h(x)$ at $x = 1$.

Let $f(x) = x^{1/3}$. Use a calculator to calculate the average rate of change of the function over each interval in the table below. What do these results imply about the slope of the tangent line to $f(x)$ at $x = 0$? Confirm your conclusions by graphing the function using a graphing calculator.

Interval	Average Rate of Change		Interval	Average Rate of Change
[-0.01, 0]			[0, 0.01]
[-0.0001, 0]			[0, 0.0001]
[-0.000001, 0]			[0, 0.000001]

Let $g(x) = |x|$. Use a calculator to calculate the average rate of change of the function over each interval in the table below. What do these results imply about the slope of the tangent line to $g(x)$ at $x = 0$? Confirm your conclusions by graphing the function using a graphing calculator.

Interval	Average Rate of Change		Interval	Average Rate of Change
[-0.01, 0]			[0, 0.01]
[-0.0001, 0]			[0, 0.0001]
[-0.000001, 0]			[0, 0.000001]

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