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### 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. $$a = 3$$
Solution
1. $$a = 1$$
2. $$a = 4$$
Solution

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

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

1. Solution
2. Solution
3. 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
4. What is the equation of the tangent line to $$f\left( x \right) = 3 - 14x$$ at $$x = 8$$. Solution
5. 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
6. What is the equation of the tangent line to $$\displaystyle f\left( x \right) = \frac{5}{x}$$ at $$\displaystyle x = \frac{1}{2}$$? Solution
7. Determine where, if anywhere, the function $$g\left( x \right) = {x^3} - 2{x^2} + x - 1$$ stops changing. Solution
8. 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
9. 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