Math 1431 Final Exam Review - UH

Math 1431 Final Exam Review 1

Math 1431

Final Exam Review

Comprehensive exam. I recommend you study all past reviews and practice exams as

well. Know all rules/formulas.

Make a reservation for the final exam. If you miss it, go back and re-schedule. Best

times to find one is 9 p.m. – midnight.

Do the Teacher Evaluations on CASA, +1 point on the final exam grade.

Practice Final Exam is for practice and extra credit; 5% of the highest score is added to

the final exam grade.

The final exam raw score will replace one missed test or lowest test grade (if it’s better).


1. Find the following limits (if they exist):

a. 5

lim ( 1)x

x x

b.

21

1lim

1x x

c. 2

1

1lim

1x

x

x

d. 3

0

2

31

lim1

4x

x

x

e. 0

1 1lim

ln( 1)x x x


Recall: 0

sin( )limx

ax a

bx b and

0

1 cos( )lim 0x

ax

bx

f. 0

4lim

sin(10 )x

x

x

g. 0

1 2lim

6x

cos x

sin x

2. Use the following graph to find the following limits, if they exist.

a. 1

limx

f x

b. 1

limx

f x

c. 1

limx

f x

What type of discontinuity occurs at x = 1? at x = 0? at x = -3?


3. Describe any discontinuities of the given function.

2

5 3

9( ) 3 3

3

10 3

x

xf x x

x

x

Step 1: Is the function defined at the x-value?

Step 2: The limit must exist.

Step 3: lim ( )ax

f x

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

Hence, at x = f(x) is: CONTINUOUS or DISCONTINUOUS

Type: JUMP REMOVABLE INFINITE

Hence, at x = f(x) is: CONTINUOUS or DISCONTINUOUS

Type: JUMP REMOVABLE INFINITE


5. Find values of A and B so that the function is continuous:

23 1 4

f 4

1 4

x x

x A x

Bx x

Find f(4).

Find each one-sided limit:

4

4

lim ( )

and

lim ( )

x

x

f x

f x

These must be equal!


Know the definition of the derivative.

0

f ff ' lim

h

x h xx

h

, provided the limit exists.

6. Use the limit definition of the derivative to find '( )f x for ( ) 1f x x .

Then find '(1)f .

7. Find the derivative of the following:

a. f 8cotx x Then find f '2

.


b. 3

f sin sinh( 2 )x x x

c. f 3 tan 2 arctan( ) 4arcsin(4 )x x x x x

d. 2

2 3

1f

( 1)

xx

x

Then find

0x

df

dx

.


e. 5 3 4f 4 cosh ( ) ln( 3 )xx x x x

f. 2( ) (log )xef x x

Know how to find a second derivative of all types of functions.


8. Given 2 6 2y xy . Compute

(6,4)

dy

dxand give the equations of the tangent and normal

lines at that point.

9. Find f 1 2 given 1 1

1 2, (3) 1, ' 1 , ' 3 2, ' 24 6

f f f f f .

Recall: 1 1'( )

'( )f b

f a


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

Find the following values.

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

. h' h

. h' h

g x

2

a 1 if x e

b 2 if x f x 2g x


11. Use the intermediate value theorem to show that the function 3( ) 2 1f x x has a root on

the interval 2,1 .

12. Verify that the Rolle’s Theorem applies to the function 2f x cos x over 0, . Then

find all points in this interval that satisfy Rolle’s Theorem.

If f continuous on 0, ?

If f differentiable on 0, ?

f(0) = f( ) =

Recall: Suppose that f is continuous on the closed interval a,b and differentiable on the

open interval a,b . If 0f a f b , then there is at least one number c in a,b for

which 0f ' c .

Make sure to review the mean value theorem (Section 3.2).


13. Use the guide to curve sketching to sketch 34 4)( xxxf .

Domain of ( )f x : Asymptotes:

x-intercept(s): y-intercept(s):

3 2'( ) 4 12f x x x Find when '( ) 0f x : Find when '( )f x is undefined:

Critical Points:

( )f x :

'( )f x :

2''( ) 12 24f x x x Find when ''( ) 0f x : Find when ''( )f x is undefined:

( )f x :

''( )f x :


14. The following graph is the graph of ''( )f x of a polynomial function f . Determine where f

is concave up and concave down.

a. f is concave up on: b. f is concave down on:

15. The value x = a is a critical number for f (x). Use the second derivative test to classify it as a

local maximum, local minimum or neither.

3 2f 12 , 8x x x x


16. Find the absolute extremum of the function 2 4( ) xf x e on [-2, 2].

17. Use differentials to approximate 63 .

Recall: 'f x h f x f x h


Recall: x

a

dF' x f t dt f x

dx and using the FTOC

b

a

f x dx F b F a , where

F is the antiderivative.

18. Evaluate 22 1

2

3 4x

dt dt

dx

.

19. Compute the upper sums for the function 2f x x on the interval [0, 2] associated with the

partition P = {0, 1, 2}.


20. Given the graph of f (x) below with the area of region A equal to 7/3, region B is 34/3 and

region C is 7/3.

a. Find the area of the region bounded by f (x) and the x-axis between –2 and 4.

b. Find 4

2

f x dx

.

21. Integrate:

a. 523 3 33x x x dx

b. 3 2

3

2x xdx

x


c. 4

2 1x x dx

d. 2

0

3cos 2

2x dx

e. 2

11 xxe dx


f. 5sinh 10x dx

g. csc 2 cot 2x x dx

h. 5sin cosx x dx


i.

4

2

2

3

xdx

x

j. ln 2

0

sinh( )

cosh( ) 1

xdx

x


k. 2

1

1dx

x x

l. 2 4x dx

22. Suppose the distance, in feet, covered by a car moving along a straight road t seconds after

starting from rest is given by 2( ) 2 48f t t t . Find:

a. the position at t = 5.

b. the velocity at t = 7.

c. the acceleration at t = 7.


23. A ball is thrown straight up with an initial velocity of 128 ft/sec, so that its height (in feet)

after t seconds is given by 2( ) 16 128f t t t . What is the instantaneous velocity at time t = 5,

and is the ball rising or falling at this time?

24. Suppose a spherical balloon is inflated at the rate of 10 cubic centimeters per minute. How

fast is the radius increasing when the radius is 5 centimeters?

Picture:

Given: Question:

Relation:

Review all Related Rates problems (Section 3.1) from the class notes, quizzes, practice tests!


25. Find two numbers whose sum is 10 and the sum of their squares is a minimum.

Review all Optimization Problems (Section 5.1) from the class notes, quizzes, practice tests!


Recall:

Removable Discontinuity occurs when:

lim ( )x c

f x

f(c)

Jump Discontinuity occurs when:

lim ( )x c

f x

and lim ( )x c

f x

exists, but are not equal

Infinite Discontinuity occurs when:

lim ( )x c

f x

on at least one side of c. Infinite discontinuities are generally associated

with having a vertical asymptote.

Determine if the following function is continuous or discontinuous at x = -2 and x = 3.

2

2, 2, 3

6( )

1, 2, 3

5

xx

x xf x

x


Find all points of discontinuity and classify them for:

2 1, 0

( ) 1, 0 2

1, 2

6

x x

f x x x

xx

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Math 1431 Final Exam Review - UH