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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.

**Give exact answers**

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.