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- Evaluate each of the following limits or state it does not exist. If it does not exist, give a reason. (a) lim x 2 + 7x - 8 %~1 x - 1 (b) lim (1 + 2x)1/% %~w
(c) limf(x) where x~ x. (x. + 1)
(d) lim
~~ -1-
lim~
.~o .:.. 3
(e)
- Ifjis a differentiable function and y =
- Given thatf^ and^ aare differentiable functions, f(4) and h(x) = M. Find h/(4). a(x)
4.^ If fix)^ = sin^ (X3), fmd fl(X).
- Iff(x) = COS(cos (Jrx», find n1.1).
Find an equation of the line tangent to the graph of the equation x 2 ,6.
(0,6).
- FindthemaximumvalueofJ<x) = x3 +2x2 -4xontheinterval[-3, 1].
The acceleration due to gravity on Mars is 3.72 m/ sec2. If a rock is thrown straight up from the surfaceof Mars with an initial velocity of 23 m/ sec, how high will the rock go before it starts to fall?
STUDY GUIDE FOR FINAL EXAMINATION MATH 1501
{ X 2 ,
f(x) = 9 - x
x~ x>
for for
(X2) 'f(x), fmd ~.
ax
0,j/(4)
I
- (^) 9x)' = 36 at the point
9. Find the maximum value and minimum value of
3x + 6
- Let f(x) =. (^) x - 1
Find the intervals on which the graph is increasing and decreasing. Find any local extrema. Find the intervals on which the graph is concaveup and concavedown.
(a) (b) (c) (d) (e)
Find any inflection points. Find all asymptotes.
- Repeatproblem 10 for fix) = xS - 5x.
- A sphericalballoon is being blown up at a rate of 100 cm3/min. At what rate is its radius r changing when r is 4?
- Given that y is defined implicitly as a function of x by the equation x
- (^) Find the value c whose existence is guaranteedby the Mean Value Theorem for
[0, ~}. (Use your calculator to approximate the value for c.)
f(x) (x^ 3)2/3 on [0, 4].
yt(.ifJ.
= cos(xy),find
f(x) = sin (3x)^ on
Use the Fundamental Theorem of Calculus to find p' (x). j %+ (a) P(x) == 1 'v't dt
(b) P(x) == j^1 3%(t2 - 3t + 3)dt
(c) P(x) == j 20 sint2dt
% -3%
(a)(a)^ F(x)^ =^.
(b) F(x) =
smt^.^2 dt (c) F(x)
If J(x) = 3x2 J~3---=-4,fmd the averagevalue off on [2,5].
- A rectangular box is to be made by cutting out equal squaresfrom each corner of a piece of cardboard 10 inches by 16 inches and then folding up the sides. What must be the length of the side of the square cut out if the volume is to be maximized? What is the maximum volume?
21. Find the derivativeof
- (a) 9 (b) 1 (c) does not exist (d) - ~ (e) 9
2. -^ d)' = x 2 I^ I(x) + 2xJ<x)
dx
3. h/(4) = 8
- I^ I^ (x) = 3x 2 cas(x 3 )
3t + 3)dt
=^ 5sin+(x3^ - 3x1.
.Y
Answers
7.^ 8 whenx = - 2. 8. 71.1matt=6.183sec
3
9. max: /9 at x = 0
min: 0 at x = 3
- (a) dec: (-00,1) U (1,00) (b) no local extrema (c) concaveup: (1,00) concavedown: (-00,1) (d) no inflection points
(e) hor asym: y = 3
ver asym: x = 1
11. (a) inc: (-00,0) U (4,00)
dec: (0, 4)
(b) loc max: 0 at x =
loc min: -256 at x = 4
(c) con up: (3, 00)
con down: (-00, 3)
(d)
(e)
no asymptotes
25 161t"
::=0.4974 cm /min
-2 1l"~ -5.
- c ~ 0.
1531l" 5
--^99 1l" 4 ~~
(b)
(c)
(d)
1 .
- (a) + c
(b) 3 4)3/
(c)
--COg^1 6 (X 3 + 5 ) + C 18
- (a) 2 (b) 1 (c) 1/ (d) 5/
P^ , (x)^ = 4r:yX P (X)^ ,^ = 27 X 2 - 27 X + 9 P'(X) =(2x - 3) sin (x2 - 3x)
- (a) (b) (c)
- 294
- squarecut-out: 2 in; volume: 144 in
22. (60x - 120x)sm(x-z. 3 3 x )cos(x - 3 3 3 x) 3
