2nd Quarterly Review
AP Calculus AB
3.7 Optimization, 4.1-4.6 Integration, 5.1-5.5 Natural Logarithms and Exponential Functions
GRAPHING CALCULATOR MAY BE REQUIRD FOR SOME PROBLEMS.
ROUND TO 3 DECIMAL PLACES WHEN APPROXIMATING A VALUE.
1.
A ball is thrown straight up in the air from ground level. Its height in feet after t seconds is given by
s(t)  16t 2  50t . When does the ball reach its maximum height? What is its maximum height?
2.
A farmer has 2000 feet of fencing to enclose a pasture area. The field will be in the shape of a rectangle
and will be placed against a river where there is no fencing needed. What is the largest area field that
can be created and what are its dimensions?
3.
A box with no top is folded from a piece of cardboard that is 12” by 15”. What are the dimensions of the
box with the greatest volume that can be created from that piece of cardboard?
4.
Given the function y  
5.
Determine the indefinite integral:
6.
Determine the indefinite integral:
7.
Solve the differential equation:
8.
1 2
x  5 , what is the largest rectangle that can be inscribed under the curve
4
with its base positioned on the x-axis? Assume the upper corners of the rectangle intersect the curve.
 (6u  10u - 2) du
 csc x(cot x  csc x) dx
2
dQ
 14t 6 Q(2)  9
dt
Use left endpoints and 4 rectangles to find the approximation of the area of the region between the
graph of the function 6x 2  x 1 and the x-axis over the interval [6, 14].
9.
Use upper, lower, midpoint and trapezoid sums to approximate the area of the region bounded by the
function f (x)  2e3x , the x –axis, y-axis and x = 1, using 4 subintervals.
10.
Evaluate the integral:
3
 2s
3
ds . Give your answer in the form of a fraction.
0
11.
Sketch the region whose area is given by the definite integral and then use a geometric formula to
5
evaluate the integral.
 5  x  dx
0
4
12.
Evaluate the definite integral:
u
3/2
du . Give your answer in the form of a fraction.
0
4
13.
Evaluate the definite integral of the trig function.
 (5 sin x  2 cos x) dx
0
5
14.
Evaluate the definite integral of the function.
 (2x  5 cos x) dx
0
15.
Determine the area of the region bounded by the curve y  2x  2sin x , the lines x = 0, x =  and the xaxis.
16.
17.
Find the average value of the function f (x)  24 12x 2 over the interval [-5,5].
Determine all values of c guaranteed by the Mean Value Theorem for Integrals in the interval [1,3] for
5 x2  4
the function f (x) 
.
x2
18.
Find F '(x) given F(x) 


x2
 (12t  1) dt
5
2 x3
19.
Find F '(x) given F(x) 
 (t
4
) dt
3
Determine the indefinite and definite integrals.
20.
21.
22.
23.
24.
25.
26.
 x 5 x
 x dx
6
4
7
dx
5
5x 2  7x  3
dx

x
3x
 x 2  2 5 dx


 cos 4 x dx
 4 x cos x  dx
3
4
cos x
dx
6
x
 sin
4 
27.

x cos(2x 2 ) dx
0
Differentiate.
28.
 17x 
f (x)  ln  2
 x  10 
29.
f (x)  ln(ln 3x5 )
30.
y
31.
y  xx
32.
f (x)  x e
33.
f (x)  3x 4 g4 5 x
34.
f (x) 
35.
f (x)   x
36.
xey  2x  3y  5
x 5 5x  7
(x  1)2
(Use logarithmic differentiation.)
(Use logarithmic differentiation.)
3 x
ex
ln x
(Use implicit differentiation to find
dy
.)
dx
Evaluate the indefinite and definite integrals.
37.
 tan 5 d
5
38.
1  cos 3 
 3  sin 3  d
1
39.
 7xe
e
40.

5 x 2
dx
5  ln x 4
x
1
dx
10 x 7 dx
41.
 ln e
42.
25
dx
x (1 x)
2x  4
 x 2  4x  29 dx
e tan x
 cos2 x dx
43.
44.
9
2
45.
e
46.
e x
 4  e x dx
e3
47.
x
4  ex dx
1
 x dx
1
48.
49.
50.
 5 csc 3x dx
 x cot 3x  dx
 sec x  sec x  dx
2
2
Review of Exponents and Logarithms
Solve for x.
1
51.
32 x3 
3
3x
 1
2 x3
7
 
52.
 49 
log 4 x  log 4 (x  12)  3
53.
54.
ln x  ln(x 1)  ln(x  7)
55.
 x 3 5x  7 
ln
 7y 3 
Use properties of logs to expand the following expression:


5
Applications.
56.
Determine the area of the region bounded by the function y 
1
4  x2
, the x-axis, the y-axis and the
line x = 1.
58.
dy
7x
.

dx 3  x 2
Use Calculus techniques to determine the relative extrema of y  ln(x2  4x  6) .
59.
Determine the equation of the tangent line to y  4x 2  ln x at the point (1,4).
60.
Use Calculus techniques to find the extrema and points of inflection for the function y  x2 e x .
57.
Use integration to find a general solution to the differential equation