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Math 20A Calculus 1 Exam #2

Northup Summer 2020

Show all work and use correct notation. Both are required for full credit for all problems.
Expressions should be algebraically simplified.

1. Find the derivative of each of the following functions. (18 points)

a. 3 4 2)
3

( 5f xx x
x

− +=

b. ( ) 2xg x =

c.
2

1
( )

x
h x

x
=

+

d. ( ) sinxF x e x=

e. ( ) ln(sec )G x x=

f. ( )cosy x=

Math 20A Calculus 1 Exam #2

Northup Summer 2020

2. Prove that [ ] 2cot cscd xx
dx

= − using the quotient rule. (10 points)

3. Find
0

sin 6 1 cos 6
lim

3 2x
x x

x x→
− 

+ 
 

. You must show work – reading the answer from a graph is not

sufficient. (10 points)

Math 20A Calculus 1 Exam #2

Northup Summer 2020

4. Find the equation of the line tangent to the curve 2 3xy y x+ = at the point ( )4, 2 . (15 points)

5. Find the derivative of sin xy x= . Show all work. (5 points)

Math 20A Calculus 1 Exam #2

Northup Summer 2020

6. An object moves so that its height, in meters, above the ground at t seconds is given by the
function 2 4 2( ) 4 , 0f tt tt = − ≤ ≤ . (15 points)

a. Find the velocity and acceleration functions.

b. During which time interval is the object moving upwards? How do you know?

c. Find any times in the interval (0, 2) when the acceleration of the particle is 0.

7. Suppose a population is growing exponentially so that it doubles every 10 years. Find the
relative growth rate, and then predict the population in 15 years if the population at year 0 is 500.
Round to the nearest whole number. (5 points)

Math 20A Calculus 1 Exam #2

Northup Summer 2020

8. A spherical balloon is inflated at a rate of 10 cubic inches per minute. How fast is the radius of
the balloon increasing when the radius is 5 inches? Give an exact answer and an approximation
accurate to the nearest thousandth. (10 points)

9. Find the linearization ( )L x of the function ( ) 5f x x= + at 4x = . Then, use ( )L x to
approximate 10 . Round your answer to 3 decimal places. (10 points)

10. If
2xy e−= , what is dy ? (2 points)

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