math 1431 final exam review - uh
TRANSCRIPT
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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).
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Math 1431 Final Exam Review 2
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
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Math 1431 Final Exam Review 3
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?
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Math 1431 Final Exam Review 4
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
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Math 1431 Final Exam Review 5
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!
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Math 1431 Final Exam Review 6
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
.
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Math 1431 Final Exam Review 7
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
.
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Math 1431 Final Exam Review 8
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.
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Math 1431 Final Exam Review 9
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
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Math 1431 Final Exam Review 10
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
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Math 1431 Final Exam Review 11
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).
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Math 1431 Final Exam Review 12
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 :
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Math 1431 Final Exam Review 13
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
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Math 1431 Final Exam Review 14
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
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Math 1431 Final Exam Review 15
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}.
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Math 1431 Final Exam Review 16
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
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Math 1431 Final Exam Review 17
c. 4
2 1x x dx
d. 2
0
3cos 2
2x dx
e. 2
11 xxe dx
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Math 1431 Final Exam Review 18
f. 5sinh 10x dx
g. csc 2 cot 2x x dx
h. 5sin cosx x dx
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Math 1431 Final Exam Review 19
i.
4
2
2
3
xdx
x
j. ln 2
0
sinh( )
cosh( ) 1
xdx
x
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Math 1431 Final Exam Review 20
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.
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Math 1431 Final Exam Review 21
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!
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Math 1431 Final Exam Review 22
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!
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Math 1431 Final Exam Review 23
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
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Math 1431 Final Exam Review 24
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