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### Section 3-2 : Interpretation of the Derivative

For problems 1 – 3 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. $$a = - 5$$
2. $$a = 1$$
1. $$a = - 2$$
2. $$a = 3$$
1. $$a = - 3$$
2. $$a = 4$$

For problems 4 – 6 sketch the graph of a function that satisfies the given conditions.

1. $$f\left( { - 7} \right) = 5$$, $$f'\left( { - 7} \right) = - 3$$, $$f\left( 4 \right) = - 1$$, $$f'\left( 4 \right) = 1$$
2. $$f\left( 1 \right) = 2$$, $$f'\left( 1 \right) = 4$$, $$f\left( 6 \right) = 2$$, $$f'\left( 6 \right) = 3$$
3. $$f\left( { - 1} \right) = - 9$$, $$f'\left( { - 1} \right) = 0$$, $$f\left( 2 \right) = - 1$$, $$f'\left( 2 \right) = 3$$, $$f\left( 5 \right) = 4$$, $$f'\left( 5 \right) = - 1$$

For problems 7 – 9 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. Answer the following questions about the function $$g\left( z \right) = 1 + 10z - 7{z^2}$$.
1. Is the function increasing or decreasing at $$z = 0$$?
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?
2. What is the equation of the tangent line to $$f\left( x \right) = 5x - {x^3}$$ at $$x = 1$$.
3. The position of an object at any time $$t$$ is given by $$s\left( t \right) = 2{t^2} - 8t + 10$$.
1. Determine the velocity of the object at any time $$t$$.
2. Is the object moving to the right or left at $$t = 1$$?
3. Is the object moving to the right or left at $$t = 4$$?
4. Does the object ever stop moving? If so, at what time(s) does the object stop moving?
4. Does the function $$R\left( w \right) = {w^2} - 8w + 20$$ ever stop changing? If yes, at what value(s) of $$w$$ does the function stop changing?
5. Suppose that the volume of air in a balloon for $$0 \le t \le 6$$is given by$$V\left( t \right) = 6t - {t^2}$$ .
1. Is the volume of air increasing or decreasing at $$t = 2$$?
2. Is the volume of air increasing or decreasing at $$t = 5$$?
3. Does the volume of air ever stop changing? If yes, at what times(s) does the volume stop changing?
6. What is the equation of the tangent line to $$f\left( x \right) = 5x + 7$$ at $$x = - 4$$?
7. Answer the following questions about the function $$Z\left( x \right) = 2{x^3} - {x^2} - x$$.
1. Is the function increasing or decreasing at $$x = - 1$$?
2. Is the function increasing or decreasing at $$x = 2$$?
3. Does the function ever stop changing? If yes, at what value(s) of $$x$$ does the function stop changing?
8. Determine if the function$$V\left( t \right) = \sqrt {14 + 3t}$$ increasing or decreasing at the given points.
1. $$t = 0$$
2. $$t = 5$$
3. $$t = 100$$
9. Suppose that the volume of water in a tank for $$t \ge 0$$ is given by $$\displaystyle Q\left( t \right) = \frac{{{t^2}}}{{t + 2}}$$.
1. Is the volume of water increasing or decreasing at $$t = 0$$?
2. Is the volume of water increasing or decreasing at $$t = 3$$?
3. Does the volume of water ever stop changing? If so, at what times(s) does the volume stop changing?
10. What is the equation of the tangent line to $$g\left( x \right) = 10$$ at $$x = 16$$?
11. The position of an object at any time $$t$$ is given by $$Q\left( t \right) = \sqrt {1 + 4t}$$.
1. Determine the velocity of the object at any time $$t$$.
2. Does the object ever stop moving? If so, at what time(s) does the object stop moving?
12. Does the function $$Y\left( t \right) = 2{t^3} + 9t + 5$$ ever stop changing? If yes, at what value(s) of $$t$$ does the function stop changing?