# Calculus I, Exam #3 Fall 2005 Primarily Chapters 4 and 5

Calculus I, Exam #3 Primarily Chapters 4 and 5 Fall 2005 Name____________________________ Please show all work to receive full credit. Answers without appropriate supporting work will not receive credit. Each problem is worth 10 points.

unless indicated otherwise. 1.

Evaluate the integrals a.  3  4

x

3 2

dx x

b.  tan

x

sec 2

xdx

2.

Evaluate  5

x x

2   16

dx

9

3. Use the limit process to find the area of the region between

f

(

x

)  2

x

2  1 and the x-axis over the interval [0,1]. 

x

c i

 4 Evaluate: a. 2  2

x

 2

dx d

b.

dx x

0  

t

2  2

t

 4

dt

5.

A certain type of bacteria increases continuously at a rate proportional to the number present. If there are 500 present at a given time and 1000 present 2 hours later, how many will there be 5 hours from the initial time given? (Find the growth function and show all work) (Round your answer to 3 decimal places.)

6.

a. Solve the separable differential equation

y

x

 1  

y

'  0 . Leave your answer in implicit form. b. Find the particular solution of the differential equation above that satisfies the initial condition

y

(  2 )  1 7.

Solve the first-order linear differential equation

y

'  3

x

2

y

e x

3 8.

a. Evaluate the following 

x x

 2

dx dy

b. Find

dx

for

y

x

cosh

x

 sinh

x

9.

a. Find the average value of

f

(

x

)  sin

x

on the interval    4 ,  2   b.   3 1 1 3 3 4  9

x

2

dx

10.

a. Use the definition of derivative to find

f

' (

x

) for

f

(

x

) 

x

1  1 b.

A balloon rises at the rate of 8 feet per second from a point on the ground 60 feet from an observer. Find the rate of change of the angle of elevation when the balloon is 25 feet above the ground.