calculus II review questions.

Review for the Third Midterm of Math 150 B 11/24/2014

Problem 1 Recall that 1

1−x = ∑∞

n=0 x n for |x| < 1.

Find a power series representation for the following functions and state the radius of convergence for the power series a) f(x) = x

2

(1+x)2 .

b) f(x) = 2 1+4x2

.

c) f(x) = x 4

2−x. d) f(x) = x

1+x2 .

e) f(x) = 1 6+x

.

f) f(x) = x 2

27−x3 .

Problem 2 Find a Taylor series with a = 0 for the given function and state the radius of conver- gence. You may use either the direct method (definition of a Taylor series) or known series. a) f(x) = ln(1 + x) b) f(x) = sin x

x c) f(x) = x sin(3x).

Problem 3 Find the radius of convergence and interval of convergence for the series

∑∞ n=1

(x+2)n

n4n .

Ans. Radius r=2, √ 2 − 2 < x <

√ 2 + 2. Problem 4

Find the interval of convergence of the following power series. You must justify your

answers. ∑∞

n=0 n2(x+4)n

23n .

Ans. −12 < x < 4.

Problem 5 For the function f(x) = 1/

√ x, find the fourth order Taylor polynomial with a=1.

Problem 6 A curve has the parametric equations x = cos t, y = 1 + sin t, 0 ≤ t ≤ 2π a) Find dy

dx when t = π

4 .

b) Find the equation of the line tangent to the curve at t = π/4. Write it in y = mx+b form. c) Eliminate the parameter t to find a cartesian (x, y) equation of the curve. d) Using (c), or otherwise identify the curve. Problem 7 State whether the given series converges or diverges a)

∑∞ n=0 (−1)

n+1 n22n

n! .

b) ∑∞

n=0 n(−3)n 4n−1

.

c) ∑∞

n=1 sin n

2n2+n .

Problem 8

1

Approximate the value of the integral ∫ 1 0 e−x

2 dx with an error no greater than 5×10−4.

Ans. ∫ 1 0 e−x

2 dx = 1 − 1

3 + 1

5.2! − 1

7.3! + ... +

(−1)n (2n+1)n!

+ .... n ≥ 5, for n=5

∫ 1 0 e−x

2 dx ≈ 1 − 1

3 + 1

5.2! − 1

7.3! + 1

9.4! − 1

11.5! ≈ 0.747.

Problem 9 Find the radius of convergence for the series

∑∞ n=1

nn(x−2)2n n!

.

Ans. R = 1√ e .

Problem 10 Let f(x) =

∑∞ n=0

(x−1)n n2+1

. a) Calculate the domain of f. b) Calculate f ′(x). c) Calculate the domain of f ′. Problem 11 Let f(x) =

∑∞ n=0

cos n n!

xn. a) Calculate the domain of f. b) Calculate f ′(x). c) Calculate

∫ f(x)dx.

Problem 12 Using properties of series, known Maclaurin expansions of familiar functions and their arithmetic, calculate Maclaurin series for the following. a) ex

2

b) sin 2x c)

∫ x5 sin xdx

d) cos x−1 x2

e) d((x+1) tan−1(x))

dx

Problem 13 Calculate the Taylor polynomial T5(x), expanded at a=0, for f(x) =

∫ x 0 ln |sect + tan t|dt.

Ans. T5(x) = x2

2 + x

4

4! .

Problem 14 Suppose we only consider |x| ≤ 0.8. Find the best upper bound or maximum value you can for∣∣∣sin x − (x − x33! + x55! )∣∣∣ Same question: If

( x − x

3

3! + x

5

5!

) is used to approximate sin x for |x| ≤ 0.8. What is

the maximum error? Explain what method you are using. Problem 15 The Taylor polynomial T5(x) of degree 5 for (4 + x)

3/2 is (4 + x)3/2 ≈ 8 + 3x + 3

16 x2 − 1

128 x3 + 3

4096 x4 − 3

32768 x5.

a) Use this polynomial to find Taylor polynomials for (4 + x)1/2 and (4 + x)5/2. b) Also replace x by 4x to find a polynomial for (1 + x)3/2.

Problem 16 With regard to the power series

∑∞ n=1 (−1)

n+1n− 1 4 (2x)n.

2

a) Does this series converge or not at x = 1 2 ? Why so?

b) Does this series converge or not at x = −1 2 ? Why so?

c) Determine the radius of convergence of this series.

Problem 17 a) Find the arc length function for the curve y = x2 − 1

8 ln x starting at the point

(1, 1).

b) Find the arc length for the curve y = x 5

6 + 1

10x3 over the interval [1, 2].

Problem 18 Evaluate the arc length of the curve y = xe−x

2 over the interval [0, 1]. (Do not eval-

uate the integral).

Problem 19 Set up the arc length of the ellipse x

2

a2 + y

2

b2 = 1 as an integral. (Do not evaluate the

integral).

Problem 20 a) Find the Taylor Polynomial of degree 3 centered at a=1 for f(x) =

√ x + 3.

b) Use your polynomial to approximate √ 4.8.

Problem 21 Find the equation of the tangent line to the curve x = 3 + ln t, y = t2 + 2 at the point (3, 3).

Problem 22 Find the arc length of the curve given in parametric form for −1 ≤ t ≤ 1, x = t y = t sin t

Problem 23 Find the arc length of the curve given in parametric form for 0 ≤ t ≤ π/4 x = cos 3t sin 4t, y = sin 3t cos 4t. Problem 24 Find the arc length of the curve given in parametric form for 0 ≤ t ≤ π/2 x = 5 cos t − cos 5t, y = 5 sin t − sin 5t. Ans. L=10.

Problem 25 Consider the curve parametrically as x = t3 − t, y = t4 − 5t2 + 4. a) Find dy

dx .

b) Find all the points where tangent lines are horizontal. c) Find all the points where tangent line are vertical. d) Find the slope of the tangent line at the point on the graph of the curve when t = 1.

Problem 26 Consider the power series W(x) =

∑∞ n=1

(−n)n n!

xn.

3

a) Determine the interval (open interval: you do not need to decide on the behavior at the end points) of the convergence for W(x). You may use without proof that lim→x∞(

n n+1

)n = 1 e .

b) Calculate a power series for W ′(x). c) Let I = [0, 0.1]. You are given the information, that for the secon derivative |W (2)(x)| ≤ 2 for x in I. Using the formula error term of a Taylor polynomial, show that |W(x) − x| ≤ x2 for x in I.

Problem 27 a) Find the equations of the tangent lines to the cycloid x = rθ−r sin θ, y = r−r cos θ at θ = π

2 .

b) Find all the points on the graph of the cycloid at which the tangent line is either horizontal or vertical, and find the equations of the horizontal lines.

c) Analyze the concavity of the cycloid (compute d 2y

dx2 ).

d) Find the arc length of one arch of the cycloid traced by a point on the circumfer- ence of a wheel with radius r units (i.e. θ ∈ [0, 2π].

Ans. a) y = x + r(2 − π 2 ). b) equation of tangent line y = 2r, c) d

2y dx2

= − 1 r(1−cos θ),

d) L = 8r.

Problem 28 Consider the polar curve r = θ2. a) Find parametric equations x = f(θ), y = g(θ) for this curve.

b) Find the slope of the line tangent to this curve at the point (r, θ) = (π2, π).

Problem 29 Let f(x) = cos x a) Find the Taylor polynomial of degree 4 centered at a = π for f(x).

b) Use this polynomial to approximate cos 9π 7

c) Find a bound on the error in using this polynomial to estimate f(x) if π − 1 3 ≤

x ≤ π + 1 3 .

Problem 30 Find the slope of the tangent line to the curve that satisfies r = d

1+e sin θ ,

at the point θ = π 3 , where the equation is in polar coordinates with d > 0 and

0 < e < 1.

Problem 31 a) Sketch the curve r =

√ θ, in polar coordinates, for 0 ≤ θ ≤ 2π.

Find the area enclosed by this curve.

Problem 32

4

Find a polar equation for the circle x2 + (y − 3)2 = 9.

Problem 33 Replace the following polar equations by equivalent Cartesian equations. a) r = 1 + 2r cos θ b) r = 1 − cosθ c) r = 4

2 cos θ−sin θ.

Problem 34 Find the area of the surface generated by revolving the curve C a) y = 2

√ x, 1 ≤ x ≤ 2, about the x-axis.

b) x = y 3

3 , 0 ≤ x ≤ 1, about the y-axis.

Ans. a) S = 8π 3 (3 √ 3 − 2

√ 2). b)

π( √ 8−1) 9

.

Problem 35 Find surface area of the solid generated by revolving the smooth curve C represented by the parametric equations x(t) = 2t3, y(t) = 3t2, 0 ≤ t ≤ 1, about the x-axis.

Ans. S = 24π 5 ( √ 2 + 1).

Problem 36 I) Find the rectangular coordinates of each point whose polar coordinates are: a) (4, π

3 ), b) (−2, 3π

4 ).

II) Find Four polar coordinates (2 with r > 0 and 2 with r < 0) of each point whose rectangular coordinates are: a) (4, −4), b) (−1, −

√ 3).

Problem 37 Find the arc length of the logarithmic spiral represented by r = f(θ) = e3θ from θ = 0 to θ = 2.

Ans. s = √ 10 3 (e6 − 1).

Problem 38 Find the area of the region enclosed by the the limacon r = 2 + cos θ. Ans. 9π

2 .

Problem 39 Find the area of the region that lies outside the cardioid r = 1 + cos θ and inside of the circle r = 3 cos θ. Ans. π.

5