Math 1413 Test 2 Review 1

 

Math 1431 Exam 2 Review

Material Covered: Chapters 1 and 2

Know your instructor’s name.

Know your lab’s TA name, lab’s time and room number.

Test: 50 minutes; CASA Testing reservation required.

Test is part multiple choice and part free response (with parts). You will have the multiple choice grade right away, but the free response will take some time to grade.

No formulas are provided. No calculators allowed.

You will be given scratch paper.

For the free response you’ll be given stapled pages asking for your name, lab info, etc to do all your free response work on. Show all your work here and not on the scratch paper.

Do the practice test (20 attempts allowed) for more practice and extra credit.

1. Evaluate the following limits, if they exist.

a. 0

5lim 2 1x

x xx

b. 2

31

2limx

x

x x

Math 1413 Test 2 Review 2

 

c. 2

22

3 10lim

4x

x x

x

d. 0

4 2limx

x

x

2. Find the following limits, if they exist.

Recall: 0

sin( )limx

ax a

bx b and

0

1 cos( )lim 0x

ax

bx

a. 0

sec(5 )lim

2x

x

x

b. 0

5lim

tan(9 )x

x

x

Math 1413 Test 2 Review 3

 

c. 2

0

1 cos (2 )lim

7x

x

x

d. 0

1 2lim

6x

cos x

sin x

3. The function f is given below along with its graph. Find the following limits, if they exist.

2

2 7 2

1 2 2

1 2<

x , x

f x x , x

x , x

a. 2

limx

f x

b. 2

limx

f x

c. 2

limx

f x

What type of discontinuity occurs at x = -2? at x = 2?

Math 1413 Test 2 Review 4

 

4. Find the points of discontinuity (if any). Classify each as removable, jump or infinite.

a. 3 24 1f x x x x

b. 3

28

2

xf x

x x

3 3 2 2

3 3 2 2

( )( )

( )( )

a b a b a ab b

a b a b a ab b

5. Determine if the following function is continuous at x = -2. If the function is not continuous, state the type of discontinuity.

25

27

25

)(2 xx

x

xx

xf

Step 1: Is f(-2) defined?

Step 2: Check to see if 2

lim ( )x

f x

exist. 2

lim ( )x

f x

must equal 2

lim ( )x

f x

.

Does 2

lim ( )x

f x

exist?

Step 3: Does 2

lim ( )x

f x

= f(-2)? i.e. Compare #1 and #2 above.

Hence, f(x) is: CONTINUOUS at x = -2 or DISCONTINUOUS at x = -2 Type: JUMP REMOVABLE INFINITE

Math 1413 Test 2 Review 5

 

6. Find the values of a and b such that f (x) is differentiable.

2

2

10, 2

( )

6 , 2

ax x

f x

x x b x

You must first check continuity at x = 2.

f(2)

2lim ( )x

f x

2lim ( )x

f x

Next, work on differentiability.

'f

'f

Math 1413 Test 2 Review 6

 

7. Find the derivative of the following functions using the definition.

Recall: h

xfhxfh

)()(lim

0

2 5f x x x

8. Find the derivative of the following:

a. 2

1( ) sec tan

2f x x x

x

Math 1413 Test 2 Review 7

 

b. 23( ) 3 1f x x

c. 2 3( ) 4 ( )f x x x x

d. 2

)1)(1()(

x

xxxf

Math 1413 Test 2 Review 8

 

e. )164(sin)( 24 xxxf

9. Find

102

4

3 34

3

3

xxxdx

d.

10. Find 2x

dy

dx

for 2 2f x x cos x .

11. Find dx

dyusing implicit differentiation for 2 2 7x y xy

Math 1413 Test 2 Review 9

 

12. Find the equation of the tangent line to the curve 1

xf x

tan x

at x =0.

13. Write the equation of the normal line to sin cos 0x y at the point ,4 4

.

Math 1413 Test 2 Review 10

 

14. Suppose we are given the data in the table about the functions f and g and their derivatives.

Find 2'(2) ( ) ( ) ( )h if h x f x g x .

Extra Review Questions for Lab

Find the derivative of the following functions using the definition.

Recall: h

xfhxfh

)()(lim

0

a. 2

4f x

x

b. ( ) 2g x x

x 1 2 3 4

( )f x 3 2 1 4

'( )f x 1 4 2 3

( )g x 2 1 4 3

'( )g x 4 2 3 1