Math 1431 Test 4 Review
1. Findthederivative:
a. y  ln e x  4 sinh(x) 

b. y  sin ln(5  x)6 c.
y  x 2 e 2 x  ln e 2 x d.
y  e x  cosh( 3 x ) e.
f ( x)  ln(5 x 2 )  e 6 x  arctan(5  2 x) f.
y  (tan x)( x 7) g.
f ( x)  arctan(2 x 3 ) h.
f (x)  arcsin 3x 2 i.
y  cosh(3x)  sinh(4x) 2
2
 
2. Integrate:
4e
a.
1
 x dx e
b.
 csc 2 x
9x 
  2  5 cot x  e dx c.
 sec (3x)dx d.
2

 /4
0
e. 
sec(x)tan(x)dx x2
dx x3


f.  3x 3  2 x 2  5 dx 4
g.  x dx 1
0
h.

8
1
1 x
dx 5x  2
dx x
i.

j.
 4  x2 dx 2
x
k.
 4  x2 dx l.
 1  x2 dx m.
2
 10 x sin  x  dx n.
 2
o.
x
x 2  9dx 1
6
p.
x5


 tanh x dx 1 x
0
q.
x
2
dx 6

4  9 x2
dx b
3. Compute  f (x)dx ifF’ x f x .
a
4. Giveanantiderivativeoff x cos 3x whosegraphhasy‐intercept3.
5. Compute:
d 23x
a.
sin(3t 3 )dt 
0
dx
d 1
b.
cos(2t 2  1)dt dx  2 x
d 35 x
c.
t  1 dt dx  4 x 2
x2
6. Given F(x)   (t  2) dt, find:
3
a. F( 3) b. F (2) 7. Thefunctionf(x)givenbelowiscontinuous,findaformulaforf(x):
a. 
2
x
 t  1 f  t  dt  sin x
x
f (t )
dt t2
2
b. 2 x 4  3x 2  6  
8. Thegraphsoffandgareshown.
RegionsI,II,IIIandIVhaveareas1,3,5and7respectively.
RegionsV,VI,VIIandVIIIhaveareas1,3/2,5/2and5respectively.
y = g (x)
y = f (x) I
III
2 II
4 6 IV
2
8
VIII VI V 4
6 8
VII Give:
a.
  f  x   2 g  x   dx
8
.
2
b.
f
x

g
x
dx







6
0
9. The graph of y = f (x) is shown. The region II has area 3
and

5
2
f (x)dx  2 .
Give the area of region I
I








I

10. Graphs of y = f (x) and y = g (x) are shown. The areas of regions I, II, III and IV are 2, 3, 1
4
and 2 respectively. Give
  2 f  x   3 g  x   dx
1
y = g
y = f
I







I

 


I



I



11. Give both the upper and lower Riemann sums for the function f (x) = – x3 +12 over the
interval [–2, 2] with respect to the partition P = {–2, 0, 1, 2}.
12. Give the Riemann sum for the function f (x) = 4 – x2 over the interval [–2, 2] with respect to
the partition P = {–2, –1, 0, 1, 2} using midpoints.
13. Give the Riemann sum for the function f (x) = 4 – x2 over the interval [–2, 2] with respect to
the partition
P = {–2, –1, 0, 1, 2} using left hand endpoints.
14. Givetheequationforthetangentandnormaltothecurve: f ( x)  ln(2 x  5)  e x 3 atthe
point(3,1).
15. The management of a large store has 1600 feet of fencing to enclose a rectangular storage yard
using the building as one side of the yard. If the fencing is used for the remaining 3 sides, find
the area of the largest possible yard.
16. Of all the rectangles with an area of 400 square feet, find the dimensions of the one with the
smallest perimeter.
17. Find the coordinates of the point(s) on the curve 8y = 40 – x2 that are closest to the origin.
18. Maximize the volume of a box, open at the top, which has a square base and which is
composed of 600 square inches of material. Let x represent each dimension of the base and let
y represent the height of the box.
19. Use differentials to approximate
63 .
20. Give the differential of f ( x)  x 2  3x at x = 1 with respect to the increment 1/10.
21. Estimate tan(28  ) usingdifferentials.
22. Ineachofthefollowing,determinewhetherornotL’Hopital’sRuleapplies.Ifitapplies,
statetheindeterminateformthenfindthelimit.
1 x  e x
a. lim
x 0
x2
x  ln x
b. lim
x 1 2x 2
2x
c.
 2
lim  1  x  
x
1 cos x
x 0
x2
ln(n  4)
e. lim
n
n2
d. lim
f.
lim3n  n 2
n
3

g. lim1  
n
n

2n
x2
x    ln x
h. lim
i.
j.
lim  e  1 3x
1
2x
x 
arctan(4 x)
x  0
x
lim
23. Findf x if f ''( x)  2 x  4, f '(0)  1, f (1)  5 .

Math 311 - Spring 2014 Assignment # 8 Completion Date: Friday

Solution

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