Converge or Diverge
Taylor and Maclaurin Series
AP Classroom #1
AP Classroom #2
Calculator
100
Which sequence diverges?
a) (-1)^(n+1)/n b)2^n/e^n c)n^2/e^n d) n/ln(n)
d) n/ln(n)
100
The coefficient of x^4 in the Maclaurin series f(x) = e^(-x/2) is
a) -1/24 b) 1/24 c) -1/384 d) 1/384
d) 1/384
100
If f(x)=∑ x^(2n) / n!, then f'(x) =
a) x^3/3 + x^5/(5)2! + x^7/(7)3! + x^9/(9)4! +...+x^(2n+1)/(2n+1)n!+⋯
b) x + 3x^3/2! + 5x^5/3! + 7x^7/4! +⋯+ (2n−1)x^(2n−1)/n! +⋯
c) 2 + 2x^2 + x^4 + x^6/3 +⋯+ 2x^[2(n−1)] /(n−1)! +⋯
d) 2x + 2x^3 +x^5 + x^7/3 +...+ 2nx^(2n-1)/n! +...
d) 2x + 2x^3 +x^5 + x^7/3 +...+ 2nx^(2n-1)/n! +...
100
(#1 On the AP Classroom Worksheet)
The position of a particle moving in the xy-plane is given by the vector {4t^3,y(2t)}, where y is a twice-differentiable function of t. At time t=1/2, what is the acceleration vector of the particle.
a) {3, 2y''(1)} b) {6, 4y''(1)}. c) {12, 2y''(1)} d) {12, 4y''(1)}
d) {12, 4y''(1)}
100
The position of an object moving along a path in the xy-plane is given by the parametric equations x(t)=5sin(πt) and y(t)=(2t−1)^2. The speed of the particle at time t = 0 is
a) 3.422 b) 11.708 c) 15.580 d) 16.209
d) 16.209
200
Which of the following series converges?
a) ∑ 1/n^(1/3) b) ∑ 1/n^(1/2) c) ∑ 1/(10n-1) d) ∑ 2/(n^2 - 5)
d) ∑ 2/(n^2 - 5)
200
The Taylor polynomial of degree 3 at x=0 for f(x)=(1+x)^1/2 is
a) 1 + x/2 - x^2/4 + 3x^3/8
b) 1 + x/2 - x^2/8 + x^3/16
c) 1 + x/2 - x^2/8 + x^3/8
d) 1 - x/2 + x^2/4 - 3x^3/8
b) 1 + x/2 - x^2/8 + x^3/16
200
Which of the following is the interval of convergence for the series ∑(x+2)^n/2n
a) -4 < x < 0 b) −4 ≤x< 0 c) -2 < x < 0 d) -2 ≤ x < 2
a) -4 < x < 0
200
To what number does the series ∑(−e/π)^k converge?
a) 0 b) −e/(π+e) c) π/(π+e) d) The series does not converge
c) π/(π+e)
200
Let S be the region in the first quadrant bounded above by the graph of the polar curve r = cos θ and bounded below by the graph of the polar curve r = 2θ, as shown in the figure above. The two curves intersect when θ = 0.450.
What is the are of S?
a) 0.232 b) 0.243 c) 0.271 d) 0.384
b) 0.243
300
Which of the following alternating series diverges?
a) ∑ (-1)^(n+1) (n-1)/(n+1)
b) ∑ (-1)^(n+1) / ln(n+1)
c) ∑ (-1)^(n-1) / n^(1/2)
d) ∑ (-1)^(n-1) (n)/(n^2+1)
a) ∑ (-1)^(n+1) (n-1)/(n+1)
300
The coefficent of (x - π/4)^3 in the Taylor series about π/4 of f(x) = cosx is
a) 1/12 b) -1/12 c)1/6(2)^1/2 d) -1/3(2)^1/2
c)1/6(2)^1/2
300
Which of the following statements about the series ∑1/(2^n−n) is true?
a) The series diverges by the nth term test
b) The series diverges by limit comparison to the harmonic series ∑1/n
c) The series converges by the nth term test
d) The series converges by limit comparison to the geometric series ∑(1/2)^n.
d) The series converges by limit comparison to the geometric series ∑(1/2)^n.
300
What is the radius of convergence of the Maclaurin series for 2x/(1+x^2)?
a) 1/2 b) 1 c) 2 d) infinite
b) 1
300
The velocity vector of a particle moving in the xy-plane has components given by dx/dt=sin(t^2) and dy/dt=e^cos(t). At time t = 4, the position of the particle is (2, 1).
What is the y-coordinate of the position vector at time t = 3 ?
a) 0.410 b)0.590 c) 0.851 d) 1.410
b) 0.590
400
For which of the following series does the Ratio Test fail?
a) ∑1/n! b) ∑n/2^n c) 1 + 1/2^(2/3) + 1/3^(3/2) + 1/4^(3/2) d) ∑n^n /n!
c) 1 + 1/2^(2/3) + 1/3^(3/2) + 1/4^(3/2)
400
The Taylor polynomial of degree 3 at x=1 for e^x is
a) e[1 + (x-1) + (x-1)^2 /2 + (x-1)^3 /3]
b) e[1 + (x+1) + (x+1)^2 /2! + (x+1)^3 /3!]
c) e[1 + (x-1) + (x-1)^2 /2! + (x-1)^3 /3!]
d) e[1 - (x-1) + (x-1)^2 /2! + (x-1)^3 /3!]
c) e[1 + (x-1) + (x-1)^2 /2! + (x-1)^3 /3!]
400
Which of the following is a power series expansion of e^x + e^−x / 2?
a) 1 + x^2/2! + x^4/4! + x^6/6! +⋯+ x^2n/(2n)! +⋯
b) 1 − x^2/2! + x^4/ 4! − x^6/6! +⋯+ (−1)^n x^(2n)/(2n)! +⋯
c) x + x^3/3! + x^5/5! + x^7/7! +⋯+ x^(2n+1)/(2n+1)! +⋯
d) x − x^3/3! + x^5/5!−x^7/7! +⋯+ (−1)^n x^(2n+1)/(2n+1)! +⋯
a) 1 + x^2/2! + x^4/4! + x^6/6! +⋯+ x^2n/(2n)! +⋯
400
A particle moves in the xy-plane so that its position for t ≥ 0 is given by the parametric equations x = ln (t + 1) and y = kt^2, where k is a positive constant. The line tangent to the particle’s path at the point where t = 3 has slope 8.
What is the value of k?
a) 1/192 b) 1/3 c) 4/3 d) 16/3
b)1/3
400
If the infinite series S=∑(−1)^n+1 (2/n) is approxiately by Pk=∑(−1)n+1 (2/n), what is the least value of k for which the alternating series error bound guarantees that |S−Pk|<3/100?
a)64 b) 66 c) 68 d)70
b) 66
500
The power series x +x^2 /2 + x^3 /3 +...+ x^n /n +... converges if and only if
a) -1 < x < 1 b) -1 ≤ x ≤ 1 c) -1 ≤ x < 1 d) -1 < x ≤ 1
c) -1 ≤ x < 1
500
The Taylor polynomial of degree 3 at x=0 for (x+1)^p, where p is a constant is
a) 1 +px +p(p-1)x^2 + p(p-1)(p-2)x^3
b) 1 +px +[p(p-1)/2]x^2 + [p(p-1)(p-2)/3]x^3
c) 1 +px +[p(p-1)/2!]x^2 + [p(p-1)(p-2)/3!]x^3
d) px +[p(p-1)/2!]x^2 + [p(p-1)(p-2)/3!]x^3
c) 1 +px +[p(p-1)/2!]x^2 + [p(p-1)(p-2)/3!]x^3
500
BONUS QUESTION
How many years have Me, Lucas Bauza, Josh Cohen, and Aayana Baid had Mrs Tuttle for COMBINED
10 years COMBINED
500
BONUS QUESTION
Who is Mrs Tuttle's best friend that works at the school?
Ms. Mallard
500
BONUS QUESTION
What will Mrs Tuttle's BC pass score be this year?
100%