250 STEP 4. Review the Knowledge You Need to Score High
Use a midpoint Riemann sum with three
subdivisions of equal length to find the
approximate value of
∫ 30
0
f(x)dx.
21.
∫∞
0
e−xdx
22.
∫ 0
−∞
dx
(4−x)^2
23.
∫ 1
0
lnxdx
24.
∫ 2
− 2
dx
√
4 −x^2
25.
∫ 8
− 1
dx
√ (^3) x
11.7 Cumulative Review Problems
(Calculator) indicates that calculators are
permitted.
- Evaluate limx→−∞
√
x^2 − 4
3 x− 9
.
- Find
dy
dx
atx=3ify=ln
∣∣
x^2 − 4
∣∣
.
- The graph off′, the derivative off,
− 6 ≤x≤8 is shown in Figure 11.7-1.
y
–6 –5 –4 –3 –2 –1^0123456 x
–1
1
2
3
–2
–3
78
f′
Figure 11.7-1
(a)Find all values ofxsuch that fattains
a relative maximum or a relative
minimum.
(b)Find all values ofxsuch that fis
concave upward.
(c)Find all values ofxsuch that fhas a
change of concavity.
- (Calculator) Given the equation
9 x^2 + 4 y^2 − 18 x+ 16 y=11,
find the points on the graph where the
equation has a vertical or horizontal
tangent.
30. (Calculator) Two corridors, one 6 feet
wide and another 10 feet wide meet at a
corner. (See Figure 11.7-2.) What is the
maximum length of a pipe of negligible
thickness that can be carried horizontally
around the corner?
Figure 11.7-2
- Evaluate limx→− 1
1 +cosπx
x^2 − 1
.
- Determine the speed of an object moving
along the path described byx= 3 − 2 t^2 ,
y=t^2 +1 whent=
1
2
.
33.
∫
2 x
√
x+ 3 dx