Question

1.

(a) Let f(x) = exp(x),x ∈ R. Show that f is invertible and compute the derivative of f−1(y) in terms of y. [5]

(b) ﬁnd the Taylor series and radius of convergence for g(x) = log(1+ x) about x = 0. [6]

Answer #1

Let f : R \ {1} → R be given by f(x) = 1 1 − x . (a) Prove by
induction that f (n) (x) = n! (1 − x) n for all n ∈ N. Note: f (n)
(x) denotes the n th derivative of f. You may use the usual
differentiation rules without further proof. (b) Compute the Taylor
series of f about x = 0. (You must provide justification by
relating this specific Taylor series to...

B.) Let R be the region between the curves y = x^3 , y = 0, x =
1, x = 2. Use the method of cylindrical shells to compute the
volume of the solid obtained by rotating R about the y-axis.
C.) The curve x(t) = sin (π t) y(t) = t^2 − t has two tangent
lines at the point (0, 0). List both of them. Give your answer in
the form y = mx + b ?...

let
f(x)=ln(1+2x)
a. find the taylor series expansion of f(x) with center at
x=0
b. determine the radius of convergence of this power
series
c. discuss if it is appropriate to use power series
representation of f(x) to predict the valuesof f(x) at x= 0.1, 0.9,
1.5. justify your answe

Let A, B ⊆R be intervals. Let f: A →R and g: B →R be
diﬀerentiable and such that f(A) ⊆ B. Recall that, by the Chain
Rule, the composition g◦f: A →R is diﬀerentiable as well, and the
formula
(g◦f)'(x) = g'(f(x))f'(x)
holds for all x ∈ A. Assume now that both f and g are twice
diﬀerentiable.
(a) Prove that the composition g ◦ f is twice diﬀerentiable as
well, and ﬁnd a formula for the second derivative...

The Taylor series for the function arcsin(x)arcsin(x) about
x=0x=0 is equal to
∑n=0∞(2n)!4n(n!)2(2n+1)x2n+1.∑n=0∞(2n)!4n(n!)2(2n+1)x2n+1.
For this question, recall that 0!=10!=1.
a) (6 points) What is the radius of convergence of this Taylor
series?
Write your final answer in a box.
b) (4 points) Let TT be a constant that is within the radius of
convergence you found. Write a series expansion for the following
integral, using the Taylor series that is given.
∫T0arcsin(x)dx∫0Tarcsin(x)dx
Write your final answer in a box.
c)...

Let f: [0, 1] --> R be defined by f(x) := x. Show that f is
in Riemann integration interval [0, 1] and compute the integral
from 0 to 1 of the function f using both the definition of the
integral and Riemann (Darboux) sums.

Let f : R → R be defined by f(x) = x^3 + 3x, for all x. (i)
Prove that if y > 0, then there is a solution x to the equation
f(x) = y, for some x > 0. Conclude that f(R) = R. (ii) Prove
that the function f : R → R is strictly monotone. (iii) By
(i)–(ii), denote the inverse function (f ^−1)' : R → R. Explain why
the derivative of the inverse function,...

Let f: R -> R and g: R -> R be differentiable, with g(x) ≠
0 for all x. Assume that g(x) f'(x) = f(x) g'(x) for all x. Show
that there is a real number c such that f(x) = cg(x) for all x.
(Hint: Look at f/g.)
Let g: [0, ∞) -> R, with g(x) = x2 for all x ≥ 0. Let L be
the line tangent to the graph of g that passes through the point...

Find the radius and interval of convergence for the taylor
series centered at x=0 for g(x)=x^2ln(1+x/3). Please show work.

find the taylor series f(x)=1/x at c=1 and the radius of
convergence.

ADVERTISEMENT

Get Answers For Free

Most questions answered within 1 hours.

ADVERTISEMENT

asked 5 minutes ago

asked 5 minutes ago

asked 6 minutes ago

asked 14 minutes ago

asked 20 minutes ago

asked 24 minutes ago

asked 31 minutes ago

asked 36 minutes ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago