Calculus Final Exam

Student ID: U10011024 Name: Kuan-Lun Wang

1. Evaluate the following integral.

(a)

∫ π/2

0

sinx cos3 xdx. (b)

∫ 1

0

x(ex2

+ 2)dx.

(c)

∫ π/2

π/6

cosx

1 + sinxdx. (d)

∫ −2

−3

dx√4− (x + 3)2

.

(a)

Let u = cosx, du = − sinxdx.∫ π/2

0

sinx cos3 xdx = −∫ 0

1

u3du = −[1

4u4]01 =

1

4.

(b)

Let u = x2, du = 2xdx.∫ 1

0

x(ex2

+ 2)dx =1

2

∫ 1

0

(eu + 2)du =1

2[eu + 2u]10 =

1

2e+ 1.

(c)

Let u = 1 + sin x, du = cosxdx.∫ π/2

π/6

cosx

1 + sin xdx =

∫ 2

32

du

u= [lnu]23

2= 2 ln 2− ln 3.

(d)

Let u = x+32 , du = 1

2dx.∫ −2

−3

dx√4− (x + 3)2

=1

2

∫ 12

0

du√1− u2

=1

2[arcsinx]

120 =

π

6.

2. Find the indicated derivative.

(a)d

dx[(sinx)cosx].

(b)Find g′(4) where g(x) =x4(x− 1)

(x + 2)(x2 + 1).

Calculus Final Exam 1


(c)Find H ′(2) given the H(x) =

∫ x3−4

2x

x

1 +√tdt.

(a)d

dx[(sinx)cosx] = (sinx)cosx

d

dx(cosx ln sinx) = (sinx)cosx(cosx cotx−

sinx ln sinx).

(b)

g′(x) = g(x)[ln g(x)]′ =2x5(x− 1)

(x + 2)(x2 + 1).

g′(4) = 102417 .

(c)

H ′(x) =x

1 +√x3 − 4

− x

1 +√

2x.

H ′(2) = 0.

3. Sketch the region bounded by the x-axis and the curves

y = sin 2x and y = 2 cosx with x ∈ [−π, π], and find its

area.

Let sin 2x− 2 cosx = 0, x = ±12π.


Student ID: U10011024 Name: Kuan-Lun Wang∫ π

−π|sin 2x− 2 cosx|dx

=

∫ −π2

−π(sin 2x−2 cosx)dx−

∫ π2

−π2

(sin 2x−2 cosx)dx+

∫ π

π2

(sin 2x−

2 cosx)dx

= [−1

2cos 2x−2 sinx]

−π2−π−[−1

2cos 2x−2 sinx]

π2−π

2+[−1

2cos 2x−

2 sinx]ππ2

= 8.

4. Let P (x, y) be an arbitrary point on the curve y = x2.

Express as a function of x the distance from P to the origin

and calculate the average of this distance ax x ranges from 0

to√

3.

P (x, y) = (x, x2), OP (x, y) = (√x2 + x4).

Let u = x2 + 1, du = 2xdx.

1√3

∫ √3

0

√x2 + x4dx =

√3

6

∫ 4

1

√udu =

√3

6[2

3u

32 ]41 =

7√

3

9.

5. Let f (x) = x−2/3 for x>0.

(a)Sketch the graph of f .

(b)Calculate the area of origin bounded by the graph of f

and the x-axis from x = 1 to x = b.

(c)The region in part (b) is rotated about the x-axis. Find

the volume of the resulting solid.

(d)What happens to the area of region as b → ∞? What

happens to the volume of the solid as b→∞?

(a)



(b)∫ b

1

x−23dx = [3x

13 ]b1 = 3

3√b− 3.

(c)∫ b

1

π(x−23)2dx = [3πx−

13 ]b1 = 3π − 3πb−

13 .

(d)

limb→∞

∫ b

1

x−23dx =∞.

limb→∞

∫ b

1

π(x−23)2dx = 3π.

6. Set f (x) =

∫ x

2

√1 + t2dt.

(a)Show that f has an inverse.

(b)Find (f−1)′(0).

(a)

f ′(x) =√

1 + x2>0.

(b)

f (2) = 0, f−1(0) = 2.



(f−1)′(0) =1

f ′(f−1(0))=

√5

5.

7. Set f (x) = e−x2.

(a)What is the symmetry of the graph?

(b)On what intervals does the function increase? decrease?

(c)What are the extreme values of the function?

(d)Determine the concavity of the graph and find the point

of infection.

(e)The graph has a horizontal asymptote. What is it?

(f)Sketch the graph.

(a)

f (x) = f (−x) = e−x2.

(b)

f ′(x) = −2xe−x2.

f is increase on (−∞, 0].

f is decrease on [0,∞).

(c)

Let f ′(x) = −2xe−x2, x = 0.

f (0) = 0, limb→−∞

f (x) =∞, limb→∞

f (x) = −∞.

f is ∞ on b→ −∞;

f is −∞ on b→∞.

(d)

f ′′(x) = 4x2e−x2.

f is concare up on (−∞,∞).

(e)



limx→∞

fx = 0.

y = 0.

(f)

8. Prove that 12 + 1

3 + · · · 1n< ln(n)<1 + 12 + 1

3 + · · · 1n−1 for

all integer n ≥ 2.

Let f (x) =1

x, f ′(x) =

1

x2.

f is increase on (1, n).

m(b− a) = 1n(n− 1),

1

nis min on [1, n];

M(b− a) = 12(n− 1),

1

2is max on [1, n].

1

2+

1

3+· · · 1

n< ln(n)<1+

1

2+

1

3+· · · 1

n− 1for all integer n ≥ 2.

9. Find the absolute extreme values of y = −5 coshx +

4 sinhx.

y′ = −5 sinhx + 4 coshx =−ex + 9e−x

2.

Let y′ = 0, x = ln 3.

−5 cosh ln 3 + 4 sinh ln 3 = −3.

y is −∞ on x→ −∞;



y is −∞ on x→∞.

−3 is the absolute extreme values of y = −5 coshx+4 sinhx.

10. The half-life of radium-226 is 1620 years. What percent-

age of a given amount of the radium willremaim after 500

years? How long will it take for the original amount to be

reduced by 75%?

Let P (t) = ekt is percentage of radium-226.

P (1620) = e1620k = 12P (0), k = − ln 2

1620.

P (500) = e−25 ln 281 ∼= 0.81.

Let P (t) = 75%, t = 3240− 1620 ln 3ln 2∼= 672.36.

11.

(a)Show that F (x) = x2

√a2 − x2 + a2

2 arcsin(xa), a>0 is an

antiderivative for f (x)√a2 − x2.

(b)Calculate

∫ a

−a

√a2 − x2dx and interpet your result as an

area.

(c)The circular disk x2 +y2 ≤ a2, a>0, is revolved about the

line x = a. Find the volume of the resulting solid.

(a)

F ′(x) =√a2−x22 − x2

2√a2−x2 + a2

2√a2−x2 =

√a2 − x2.

(b)∫ a

−af (x)dx = [F (x)]a−a =

a2π

2.

(c)

Let f (x) =√a2 − x2, volume is 2

∫ a

−a2π(a− x)f (x)dx.


Student ID: U10011024 Name: Kuan-Lun Wang∫ a

−a2π(a−x)f (x)dx =

∫ a

−a2πaf (x)dx−

∫ a

−a2πxf (x)dx =

2π2a3.

Volume is 4π2a3.


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