Calculus I - Interpretation of the Derivative (Practice Problems)

Section 3.2 : Interpretation of the Derivative

For problems 1 and 2 use the graph of the function, $f\left( x \right)$, estimate the value of $f'\left( a \right)$ for the given values of $a$.

1. $a = - 2$
2. $a = 3$
$The graph of $f\left( x \right)$ on $ - 3 \le x \le 4$. It is a parabola shaped curve that comes down from approximately $\left( { - 3,6} \right)$, crosses the x-axis at about $x = - 1.8$, reaches its lowest point at approximately $\left( { - 0.2, - 2.5} \right)$ and then increases, crossing the x-axis at $x = 2$ and reaching approximately $\left( {4,3.5} \right)$ at the right end.$
Solution
1. $a = 1$
2. $a = 4$
$The graph of $f\left( x \right)$ on $0 \le x \le 5$. The graph starts at approximately $\left( {0, - 5} \right)$ and increases the whole way, slowly at first and then much more steeply. It crosses the x-axis at about $x = 3.3$ and reaches approximately $\left( {5,7} \right)$ at the right end.$
Solution

For problems 3 and 4 sketch the graph of a function that satisfies the given conditions.

$f\left( 1 \right) = 3$, $f'\left( 1 \right) = 1$, $f\left( 4 \right) = 5$, $f'\left( 4 \right) = - 2$ Solution
$f\left( { - 3} \right) = 5$, $f'\left( { - 3} \right) = - 2$, $f\left( 1 \right) = 2$, $f'\left( 1 \right) = 0$, $f\left( 4 \right) = - 2$, $f'\left( 4 \right) = - 3$ Solution

For problems 5 and 6 the graph of a function, $f\left( x \right)$, is given. Use this to sketch the graph of the derivative, $f'\left( x \right)$.

$The graph of $f\left( x \right)$ on $ - 3 \le x \le 6$. Starting at approximately $\left( { - 3,4} \right)$ the graph rises to a peak at approximately $\left( { - 2,7} \right)$, then decreases to a valley at approximately $\left( {1,2} \right)$, then increases to a peak at approximately $\left( {5,12.5} \right)$ and finally decreases a little to approximately $\left( {6,9} \right)$ at the right end.$
Solution
$The graph of $f\left( x \right)$ on $ - 4 \le x \le 4$. It is made up of three straight line segments. The first runs down from $\left( { - 4,5} \right)$ to a corner at $\left( { - 1,2} \right)$, the second runs up from there to a corner at $\left( {2,4} \right)$, and the third runs back down from there to $\left( {4,0} \right)$.$
Solution
Answer the following questions about the function $W\left( z \right) = 4{z^2} - 9z$.
1. Is the function increasing or decreasing at $z = - 1$?
2. Is the function increasing or decreasing at $z = 2$?
3. Does the function ever stop changing? If yes, at what value(s) of $z$ does the function stop changing?
Solution
What is the equation of the tangent line to $f\left( x \right) = 3 - 14x$ at $x = 8$. Solution
The position of an object at any time $t$ is given by $\displaystyle s\left( t \right) = \frac{{t + 1}}{{t + 4}}$.
1. Determine the velocity of the object at any time $t$.
2. Does the object ever stop moving? If yes, at what time(s) does the object stop moving?
Solution
What is the equation of the tangent line to $\displaystyle f\left( x \right) = \frac{5}{x}$ at $\displaystyle x = \frac{1}{2}$? Solution
Determine where, if anywhere, the function $g\left( x \right) = {x^3} - 2{x^2} + x - 1$ stops changing. Solution
Determine if the function $Z\left( t \right) = \sqrt {3t - 4} $ increasing or decreasing at the given points.
1. $t = 5$
2. $t = 10$
3. $t = 300$
Solution
Suppose that the volume of water in a tank for$0 \le t \le 6$ is given by $Q\left( t \right) = 10 + 5t - {t^2}$.
1. Is the volume of water increasing or decreasing at $t = 0$?
2. Is the volume of water increasing or decreasing at $t = 6$?
3. Does the volume of water ever stop changing? If yes, at what times(s) does the volume stop changing?
Solution