IVT, MVT, Derivatives and Newton's Method
Intro to Sequences and Infinite Series
Positive Series, P-series, Geometric Series and Convergence Tests
Power Series and Taylor Series
Antiderivatives and Definite Integrals
100
What is the graphical interpretation if the Intermediate Value Theorem?
We'll show you...
100
What does the following sequence converge to? (n^2)/((n^2)+1)
1
100
How would you determine if the series SUM(n^-3.2) converges?
It's a p-series with p=3.2 > 1 so it converges.
100
What is the radius of convergence of F(3x) if F(x) is a power series with radius of convergence R = 12?
If the power series F(x) has radius of convergence R = 12, then the power series F(3x) has radius of convergence R = 12/3 = 4.
100
Is y = x a solution of the following initial value problem? dy/dx = 1, y(0) = 1
No!
200
True or False: The derivative of the product is the product of the derivatives?
FALSE
200
b_n = sqrt(4 + b_n-1) If b_0 = 0 find b_2
sqrt(6)
200
How would you determine if the series SUM(1/(2^n +√n)) converges?
Compare with the geometric series 1/(2^n)
200
Suppose that The summation of a_n(x − 6)^ n converges for x = 10. At which of the points (a)–(d) must it also converge? X=8, x=11, x=3, x=0
The given power series is centered at x = 6. Because the series converges for x = 10, the radius of convergence must be at least |10 − 6| = 4 and the series converges absolutely at least for the interval |x − 6| < 4, or 2 < x < 10.
200
Suppose that F’(x) = f (x) and G’(x) = g(x). Which of the following statements are true? Explain. (a) If f = g, then F = G. (b) If F and G differ by a constant, then f = g. (c) If f and g differ by a constant, then F = G.
(a) False. Even if f (x) = g(x), the antiderivatives F and G may differ by an additive constant. (b) True. This follows from the fact that the derivative of any constant is 0. (c) False. If the functions f and g are different, then the antiderivatives F and G differ by a linear function: F(x)−G(x) = ax + b for some constants a and b.
300
Find (f/g)'(1) if f(1)=f'(1)=g(1)=2 and g'(1)=4
-1
300
What is the sum of the following infinite series: 1/4 + 1/8 + 1/16 + 1/32 + 1/64 + ...
1/2 (It's geometric with c=1/4 and r=1/2)
300
Linda argues that SUM(((−1)^n) * √n) is alternating and thus converges. Is this correct?
No, the terms do not converge. "Divergence Test"
300
The power series F(x) = SUM(nx^n) has radius of convergence R = 1. What is the power series expansion of F’(x) and what is its radius of convergence?
F’(x) = SUM((n^2)*(x^(n−1))) Radius of Convergence =1 (using ratio tests)
300
Explain why L_100 ≥ R_100 for f(x) = x^−2 on [3, 7].
On [3, 7], the function f (x) = x^−2 is a decreasing function; hence, for any subinterval of [3, 7], the function value at the left endpoint is larger than the function value at the right endpoint. Consequently, L_100 must be larger than R_100.
400
How many iterations of Newton's Method are required to find a root if f(x) is a linear function?
1
400
Give an example of a divergent infinite series whose general term tends to zero.
e.g. SUM(1/sqrt(n))
400
In some detail, summarize the Ratio Test.
Let's go over it...
400
Determine f (0) and f’(0) for a function f (x) with Maclaurin series T (x) = 3 + 2x + 12x^2 + 5x^3 +・ ・ ・
f (0) = 3 and f’(0)/3! = 5. Third term maclaurin is (f’’’(0)/3! ) * x^3 f(0) = 30
400
Are the following pair equal? SUM[j=1 to 4] (j^ 2) SUM[k=2 to 5] (k^2)
These two sums are not the same; the second squares the numbers two through five while the first squares the numbers one through four.
500
If f(2)=3 and f(4)=9 and f is differentiable, then f has a tangent line of slope m. Find m.
3
500
a) Explain the role of partial sums in defining the sum of an infinite series. b) explain the difference between a sequence and a series
a) The sum of an infinite series is said to be the limit of the sequence of its partial sums. If this limit does not exist, the series is said to diverge. b) sequence is a set of values, series is a sum of a set of values
500
In some detail, summarize the Root Test.
Let's go over it...
500
Find the Taylor series for f (x) centered at c = 3 if f (3) = 4 and f’(x) has a Taylor expansion f’(x) = SUM(((x − 3)^n)/n)
f (x) = C + SUM(((x − 3)^ (n+1))/ (n(n + 1))) f (3) = C = 4;
500
I = Integral from 7 to 2 of f (x)dx True or False: I is the area between the graph and the x-axis over [2, 7].
False. Integral from a to b of f (x) dx is the signed area between the graph and the x-axis.
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