Calculus-Sec-3-2-Solution

3.2  Derivatives of Basic Functions, The Product and Quotient Rule

by SGLee - HSKim, YJLim

1. Find  where .

2-5. [You may have to use a Chain Rule] Find the derivative  where  is

2.

3.

4.

5.

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 .

Thus, .

Therefore, the normal line is .

8. Where is the function differentiable? Give a formula for .

,

then .

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 .

Therefore,

10. Evaluate .

Method (1) Let , and .  Then by the definition of derivative,

.

Method (2) Note that  .

So

11-12. Differentiate the following functions.

11.

12.

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 ,  .

Now,

Since , then .

Therefore, .

14-16. Find derivatives of the following functions.

14. .

15. .

16. .

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?

.

Therefore, .

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.

(b)  .

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 .

(a) .

(b) .

(c) When . Therefore, .

(d) .

(e) .

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 .

(a) .

(b)

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?

(a) .

and

(b)

.

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

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 .

Therefore, .