2-5. [You may have to use a Chain Rule] Find the derivative where is
6. Find the equation of the tangent line to the curve at .
. So the slope of the tangent line is 20.
(Since passes through .
7. The normal line to a curve at a point is the line that passes through and is perpendicular to the tangent line to at .
Find an equation of the normal line to the curve at the point .
So the slope of the tangent line is . (Since )
Then the slope of the normal line is .
Therefore, the normal line is .
8. Where is the function differentiable? Give a formula for .
So is continuous on .
then is not differentiable at .
Therefore, is differentiable on .
9. Let . Find the values of and that make differentiable everywhere.
Note that is differentiable everywhere except . We must check at .
For to be differentiable at 2, , so . And also have to be continuous at .
, since , .
10. Evaluate .
Method (1) Let , and . Then by the definition of derivative,
Method (2) Note that .
11-12. Differentiate the following functions.
13. Show that if is differentiable and satisfies the identity for all and and ,
then satisfies for all .
. Thus, either or .
Since for all and , .
Since , then .
14-16. Find derivatives of the following functions.
17. Show that the curve has no tangent line with slope .
Since is always positive, there is no such that .
So has no tangent line with slope 0.
18. A stone is thrown into a pond, creating a wave whose radius increases at the rate of meter per second.
In square meters per second, how fast is the area of the circular ripple increasing seconds after the stone hits the water?
Let radius time. . when .
19. A particle moves along a straight line with equation of motion .
(a) When is the particle moving forward?
(b) When is (the acceleration) zero?
(c) When is the particle speeding up? Slowing down?
(a) . , when .
When , the particle moves forward.
Therefore the acceleration is 0, when .
(c) When the particle is speeding up, the acceleration is positive
and when it is slowing down, the acceleration is negative.
when and when .
Therefore the particle speeds up when , and the particle slows down when .
20. A particle moves in a straight line with equation of motion ,
where is measured in seconds and in meters.
(a) What is the position of the particle at and ?
(b) Find the velocity of the particle at time .
(c) When is the particle moving forward ?
(d) Find the total distance traveled by particle on the time interval .
(e) Find the acceleration of the particle at time .
(c) When , . Therefore, .
21. The population of a bacteria colony after hours is . Find the growth rate when .
22. A cost function is given by .
(a) Find the marginal cost function.
(b) Find .
23. If a stone is thrown vertically upward with a velocity , then its height after seconds is .
(a) What is the maximum height reached by the stone?
(b) What is the velocity of the stone when it is above the ground on its way up? On its way down?
The time is when the height of the stone is 5.
The velocity of the stone is on its way up. The velocity of the stone is on its way down.
24. If is the total value of the production when there are workers in a plant, then the average productivity is
. Find . Explain why the company wants to hire more workers if ?
Since , implies .
This means the rate of productivity is larger than the average productivity .
Therefore, if the company hires more workers, then they can expect to have a better productivity.
25. Let be the population of a bacteria colony at time hours.
Find the growth rate of the bacteria after 10 hours.
26. The angular displacement of a simple pendulum is given by
with an angular amplitude , an angular frequency and a phase constant . Find .
27. Show that .
We note . Let . Then , and as , .