3.2 Derivatives of Polynomials, Exponential Functions, Trigonometric Function, The product rule
1-4. Find the derivative 
where 
is
1. 
2. 
3. 
4. 
5. Find 
where 
.
http://matrix.skku.ac.kr/cal-lab/cal-3-1-6.html
15/14*x^(1/14) + 5*e^x
6. Find the equation of the tangent line to the curve 
at 
.
http://matrix.skku.ac.kr/cal-lab/cal-3-1-7.html 
. So the slope of the tangent line is 20.
(Since 
passes through 
.
20*x-48
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 
.
http://matrix.skku.ac.kr/cal-lab/cal-3-1-8.html 
So the slope of the tangent line is 
. (Since 
)
Then the slope of the normal line is 
.
Thus, 
.
Therefore, the normal line is 
.
-x+2
8. Where is the function 
differ-entiable? 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 every where except 
.
To 
be differentiable at 
, 
, so 
.
And also 
have to be continuous at 
.
, since 
, 
. Therefore, 
10. Evaluate 
.
http://matrix.skku.ac.kr/cal-lab/cal-3-1-11.html
Method (1) Let 
, and 
. Then by the definition of derivative,
.
Method (2) Note that 
.
So
2020
11-12. Differentiate the following functions.
11. 
12. 
13. If 
satisfies the identity 
for all 
and 
and 
, then show that 
satisfies 
for all 
.
Thus, 
or 
.
Since 
for all 
and 
, 
.
Now,
Since 
, then 
.
Therefore, 
.
14-16. Find the following derivatives.
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 wave whose radius increases at the rate of 
meter per second. In square meter per second, how fast is the area of the circular ripple increasing 
seconds after the stone hits the water?
radius 
time
,
,
,
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) 
.
(b) 
, 
.
Therefore, 
.
(c) At time 
, (the acceleration) 
is zero.
In 
, the particle is speeding up, in 
, the particle is slowing down.
20. A particle moves in a straight line with equation of motion 
, where 
is measured in second 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 the 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) 
.
(b) 
.
The time is 
when the height of the stone is 5
.
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 worker if 
?
If 
, since 
, then 
.
is the rate of productivity.
This means rate of productivity 
is larger than average productivity 
.
Thus, the company wants to hire more workers.
25. Let 
be the population of bacteria colony at time 
hours. Find the growth rate of the bacteria after 10 hours.
.
26. The angular displacement 
of simple pendulum is given by 
with the angular amplitude 
, the angular frequency 
and a phase constant 
. Find 
.
27. Show that 
.
Let 
. Then 
, and as 
, 
.
Therefore,
.