math 1431 final exam review - uh
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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).
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
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?
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
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!
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
.
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
.
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.
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
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
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).
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 :
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
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
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}.
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
Math 1431 Final Exam Review 17
c. 4
2 1x x dx
d. 2
0
3cos 2
2x dx
e. 2
11 xxe dx
Math 1431 Final Exam Review 18
f. 5sinh 10x dx
g. csc 2 cot 2x x dx
h. 5sin cosx x dx
Math 1431 Final Exam Review 19
i.
4
2
2
3
xdx
x
j. ln 2
0
sinh( )
cosh( ) 1
xdx
x
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.
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!
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!
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
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