226 STEP 4. Review the Knowledge You Need to Score High
Which of the following is the best
approximation forf′(1)?
(a) f′(1)< 2
(b) 0. 5 < f′(1)< 1
(c) 1. 5 < f′(1)< 2. 5
(d) 2. 5 < f′(1)< 3. 5
(e) f′(1)> 2
- The graph of the functionf′′on the
interval [1, 8] is shown in Figure 10.6-2.
At what value(s) ofton the open interval
(1, 8), if any, does the graph of the
functionf′:
0 12 345678
f′′(t)
y
t
Figure 10.6-2
(a) have a point of inflection?
(b) have a relative maximum
or minimum?
(c) become concave upward?
- Evaluate limx→− 2
x^2 −x− 6
x^2 − 4
.
- If the position of an object is given by
x=4 sin(πt),y=t^2 − 3 t+1, find the
position of the object att=2. - Find the slope of the tangent line to the
curver=3 cosθwhenθ=
π
4
.
10.7 Solutions to Practice Problems
1.
x^6
6
+x^3 −
x^2
2
+x+C
- Rewrite:
∫
(x^1 /^2 −x−^2 )dx
=
x^3 /^2
3 / 2 −
x−^1
− 1
+C
=
2 x^3 /^2
3
+
1
x
+C.
- Letu=x^4 −10;du= 4 x^3 dxor
du
4
=x^3 dx.
Rewrite:
∫
u^5
du
4
=
1
4
∫
u^5 du
=
(
1
4
)
u^6
6
+C
=
(x^4 −10)^6
24
+C.
4.Letu=x^2 + 1 ⇒(u−1)=x^2 and
du= 2 xdxor
du
2
=xdx.
Rewrite:
∫
x^2
√
x^2 +1(xdx)
=
∫
(u−1)
√
u
du
2