ax Recall - UHmmsosa/Math1431/final review.pdf · Math 1431 Final Exam Review 5 3. Describe any discontinuities of the given function. a. 2 5 3 x fx x b. 2 54 4 xx fx x c. Is 2

Math 1431 Final Exam Review 2

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

5lim ( 1)x

x x

b. 2

1

1lim

1x

x

x

c. 3

0

2

31

lim1

4x

x

x

Recall: 0

sin( )limx

ax a

bx b and

0

1 cos( )lim 0x

ax

bx

d. 0

4lim

sin(10 )x

x

x

e. 0

1 2lim

6x

cos x

sin x


f. 1

lim(x+1) ln x

x


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

a. 2

limx

f x

b. 2

limx

f x

c. 2

limx

f x

d. What type of discontinuity occurs at:

x = -2?

x = 2?


3. Describe any discontinuities of the given function.

a. 2 5

( )3

xf x

x

b. 2 5 4

( )4

x xf x

x

c. Is 2

5 3

9( ) 3 3

310 3

x

xf x x

xx

continuous at x = -3?

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.


4. 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


Know the definition of the derivative.

0

f ff ' lim

h

x h xx

h

, provided the limit exists.

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

Then find '(1)f .


6. Find the derivative of the following:

a. f 8cotx x Then find f '2

.

b. 3f sin sinh( 2 )x x x

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


d. 2

2 3

1f

( 1)

xx

x

Then find

0x

df

dx

.

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


f. 2( ) (log )xef x x

7. 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.


8. Suppose f(x) is an invertible differentiable function and the graph of f passes through the

points (6, -1) and (-1, 2). The slope of the tangent line to the graph of f at x = -1 is 7

2.

Find 1 ' (2)f , the slope of the tangent line to the inverse of f at 2.

9. Given 2 6 2y xy . Compute (6,4)

dy

dxand give the equations of the tangent and normal

lines at that point.


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

Find:

2 ( )'(2) if ( ) ( ) f xh h x g x e

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


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. Rolle’s Theorem: 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 .

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( ) =


13. The conclusion of the Mean Value Theorem states that there exists a point c in the interval (a, b) such that the tangent line is parallel to the line passing through (a, f(a)) and (b, f(b)).

Let xxxf 3)( 3 be defined on [-1, 1]. Find c on (-1, 1) that satisfies the conclusion of the Mean Value Theorem.

If f continuous on [-1, 1]? If f differentiable on (-1, 1)?


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

Domain of ( )f x : Asymptotes:

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

Critical Points: Find when '( ) 0f x : Find when '( )f x is undefined:

( )f x :

'( )f x : Find when ''( ) 0f x : Find when ''( )f x is undefined:

( )f x :

'( )f x :


15. Determine whether the function has a vertical tangent, cusp or neither at the given value. 4

5( ) 5( 8)f x x at c = 8

Check list:

Is ( )f c is defined?

Is '( )f c is undefined?

Create a sign chart. Does the sign chart for 'f across x = c have a sign change or not?

16. Below is the graph of the derivative of a polynomial function f. Which of the following statements is/are true or false?

a. The critical numbers for f are 0, 1 and 2.

b. The function f has two minimums.


17. 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: 18. 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


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

20. Use differentials to approximate 63 .

Recall: 'f x h f x f x h


21. Two cars are moving towards the same point. The first car started from a point that is 100 miles away and it travels south at a rate of 40 mi/h. The second one started from a point 105 miles away and it travels east at a rate of 60 mi/h. At what rate is the distance between the cars changing one hour later?

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


22. 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!


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

partition P = {0, 1, 2}.

Review all Riemann sum problems as well!


24. Evaluate 22 1

2

3 4xd

t dtdx

. Then evaluate it at x = 0.

25. Given 5

1f x dx 5 and

9

1f x dx 12 , calculate

9

52f x dx .

26. 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 .


27. Integrate:

a. 523 3 33x x x dx

b. 3 2

3

2x xdx

x

c. 42 1x x dx


d. 2

0

3cos 2

2x dx

e. 2

11 xxe dx


f. cosh tanhx x dx

g. ln 2

0

sinh( )

cosh( ) 1

xdx

x


h. 2

4

1 36dx

x

i. 2 4x dx

28. Given '( ) 3sin , ( ) 1f x x f , find ( )f x .

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ax Recall - UHmmsosa/Math1431/final review.pdf · Math 1431 Final Exam Review 5 3. Describe any discontinuities of the given function. a. 2 5 3 x fx x b. 2 54 4 xx fx x c. Is 2