Math 230 Calculus II
Practice problems for Exam II
Exam II will be based on Sections 6.3*, 6.4*, 6.6, 6.8, 7.1, 7.2, 7.3, and 7.4.
1. Review all definitions from Sections 6.3*, 6.4*, 6.6, 6.8, 7.1, 7.2, 7.3, and 7.4.
2. Review all theorems from Sections 6.3*, 6.4*, 6.6, 6.8, 7.1, 7.2, 7.3, and 7.4.
3. Solve each equation for x.
(a) e3x+1 = k
(c) ln(ln x) = 1
(b) e2x − ex − 6 = 0
(d) ln(2x + 1) = 2 − ln x
Answer: (a) 31 (ln k − 1); (b) ln 3; (c) ee ; (d)
√
−1+ 1+8e2
.
4
4. Find the limit.
(a)
lim
x→(π/2)+
etan x
(b) lim− e3/(2−x)
x→2
Answer: (a) 0; (b) ∞.
5. Differentiate the function.
√
(a) f (x) = 1 + 2e3x
(c) h(x) =
x
(b) g(x) = ee
Answer: (a)
√
1 + xe−2x
(d) k(t) = sin(et ) + esin t .
3x
√ 3e
;
1+2e3x
x +x
(b) ee
; (c)
e−2x (−2x+1)
√
;
2 1+xe−2x
(d) et cos(et ) + esin t cos t.
6. Evaluate the integral
Z 5
e dx
(a)
Z
(b)
etan x sec2 x dx
−5
Z
(c)
ex (4 + ex )5 dx
Answer: (a) 10e; (b) etan x + C; (c) 61 (4 + ex )6 + C.
7. Differentiate the function.
√
(a) f (x) = ( x)x
(c) h(x) = x5 + 5x
(b) g(x) = (ln x)cos x
(d) k(x) = log5 (xex )
√
x
Answer: (a) 12 ( x)x (1 + ln x); (b) (ln x)cos x xcos
−
sin
x
ln
ln
x
; (c) 5x4 + 5x ln 5; (d)
ln x
1
8. Find the inverse function of f (x) = log10 1 +
. Answer: 10x1−1 .
x
1
x+1
.
x ln 5
9. Find the exact value of each expression.
1
−1
(a) sin
2
(c) sin−1 (sin(7π/3))
2
−1
(d) tan sin
3
(b) sin−1 (sin(π/3))
√
Answer: (a) π/6; (b) π/3; (c) π/3; (d) 2/ 5.
√
10. Prove that cos sin−1 x = 1 − x2 .
11. Simplify the expression sin tan−1 x . Answer:
√ x
.
1+x2
12. Differentiate the function.
√
(a) f (x) = x sin−1 x + 1 − x2
(c) h(θ) = tan−1 (cos θ)
(b) g(x) = cos−1 (e2x )
2x
sin θ
; (c) − 1+cos
Answer: (a) sin−1 x; (b) − √2e
2 θ.
1−e4x
13. Evaluate the integral.
Z 1/2
sin−1 x
√
(a)
dx
2
1
−
x
0
Z
1+x
(b)
dx
1 + x2
Z
π/2
(c)
0
sin x
dx
1 + cos2 x
Answer: (a) π 2 /72; (b) tan−1 x + 12 ln(1 + x2 ) + C; (c) π/4.
14. Find the limit.
cos x
+
x→(π/2) 1 − sin x
ln x
(b) lim √
x→∞
x
1 − sin x
(c) lim
x→π/2
csc x
(ln x)2
(d) lim
x→∞
x
(a)
√
(e) lim
lim
x→∞
(f) lim
x→1
xe−x/2
1
x
−
x − 1 ln x
(g) lim (1 − 2x)1/x
x→0
(h) lim x1/x
x→∞
Answer: (a) −∞; (b) 0; (c) 0; (d) 0; (e) 0; (f) 1/2; (g) 1/e2 ; (h) 1.
15. Evaluate the integral.
Z
(a)
te−3t dt
Z
(b)
Answer: (a) − 31 te−3t − 19 e−3t + C; (b) 16 x6 ln x −
16. Evaluate the integral.
2
1 6
x
36
+ C.
x5 ln x dx
Z
π/2
sin7 θ cos5 θ dθ. Answer: 1/120.
(a)
0
Z
(b)
Z
(c)
Z
(d)
Z
(e)
Z
(f)
√
sin3 ( x)
√
dx. Answer:
x
2
3
√
√
cos3 ( x) − 2 cos x + C.
t sin2 t dt. Answer: 41 t2 − 41 t sin 2t − 18 cos 2t + C.
tan x sec3 x dx. Answer:
1
3
sec3 x + C.
tan2 θ sec4 θ dθ. Answer:
1
5
tan5 θ + 31 tan3 θ + C.
1 − tan2 x
dx. Answer:
sec2 x
1
2
sin 2x + C.
17. Evaluate the integral
Z 1 √
(a)
x3 1 − x2 dx. Answer: 2/15.
0
2
√
1
3
π
√
dt.
Answer:
+
− 41 .
24
8
√
3 t2 − 1
t
2
Z
√
1
√
(c)
dx. Answer: ln( x2 + 16 + x) + C.
x2 + 16
Z √
√
√
(d)
x2 + 2x dx. Answer: 21 (x + 1) x2 + 2x − 21 ln |x + 1 + x2 + 2x| + C.
Z √
√
(e)
x 1 − x4 dx. Answer: 14 sin−1 (x2 ) + 41 x2 1 − x4 + C.
Z
(b)
18. Evaluate the integral.
Z
5x + 1
(a)
dx. Answer: 21 ln |2x + 1| + 2 ln |x − 1| + C.
(2x + 1)(x − 1)
Z 4 3
x − 2x2 − 4
(b)
dx. Answer: 76 + ln 23 .
3
2
x − 2x
3
Z 1 3
x − 4x − 10
(c)
dx. Answer: 23 + ln 32
2−x−6
x
0
Z 2
x −x−6
(d)
dx. Answer: 2 ln |x| − 21 ln(x2 + 3) − √13 tan−1 ( √x3 ) + C.
x3 + 3x
3