Question 1

Given the function $$f(x) = \frac{x^3$}{12} - x + 1$$ defined on $$[0, +\infty)$$, what is $$\lim_{x 	o +\infty} f(x)$$?

Accepted Answer

$$+\infty$$

Question 2

Compute the derivative $$f'(x)$$ for $$f(x) = \frac{x^3$}{12} - x + 1.$$

Accepted Answer

$$f'(x) = \frac{x^2$}{4} - 1$$

Question 3

For the function $$f(x) = \frac{x^3$}{12} - x + 1$$, on which interval is $$f'(x)$$ negative?

Accepted Answer

$$x \in [0, 2)$$

Question 4

How many solutions does the equation $$f(x) = 0$$ admit on $$[0, +\infty)$$ for $$f(x) = \frac{x^3$}{12} - x + 1$$?

Accepted Answer

Exactly two solutions

Question 5

Let $$\beta$$ and $$\gamma be $$the two solutions to $$f(x) = 0$$ with $$\beta \u003c \gamma. $$Which intervals do they belong to?

Accepted Answer

$$\beta \in (1, 2)$$ and $$\gamma \in (2, 3)$$

Question 6

Given the sequence $$u_0 = 1$, u$$_{n+1} = 1 + \frac{u_$$n^3$$}{12}$$, show that u$$_{n+1} - u_n = f(u_n)$$. What does this imply about the monotonicity of (u_n$$)$$ if f(u_n$$) \leq 0$$?

Accepted Answer

$$(u_n) is $$non-increasing

Question 7

Let $$F(x) = \frac{\sin x}{x}$$ and $$G(x) = \frac{1 - \cos x}{x}$$ for $$x \u003e 0. $$What is the value of $$\lim_{x 	o 0^+} F(x)$$?

Accepted Answer

$$1$$

Question 8

Which sequence describes the positive zeros of $$F(x) = \frac{\sin x}{x}$$?

Accepted Answer

$$a_k = k\pi$$ for $$k \geq 1$$

Question 9

Let $$f \in E$$ where $$E is $$the set of functions differentiable on $$[0,1]$$ with continuous derivative. Define $$I_f(x) = \$$int_0$$^1 f(t) \cos(xt) dt. $$What is the parity of $$I_f(x)$$?

Accepted Answer

Even

Question 10

Using integration by parts, which formula correctly expresses $$I_f(x)$$ for $$x \u003e 0$$?

Accepted Answer

$$I_f(x) = \frac{f(1) \sin x}{x} - \frac{1}{x} \$$int_0$$^1 f'(t) \sin(xt) dt$$

Question 11

Suppose $$f$$ satisfies $$|f(x) - f(x_0$)| \leq C|x - x_0$|^\alpha$$ for all $$x, x_0$ \in \mathbb{R}$$ with $$\alpha \u003e 1. $$What can you conclude about $$f$$?

Accepted Answer

$$f is $$constant

Question 12

Let $$f(x) = 2x + 3. $$For which values of $$\alpha$$ does $$f$$ satisfy $$|f(x) - f(x_0$)| \leq C|x - x_0$|^\alpha$$ for some $$C \u003e 0$$?

Accepted Answer

$$\alpha = 1$$

Question 13

Given $$u_{n+1} = f(u_n)$$ with $$0 \u003c C \u003c 1$$ and $$\alpha = 1 in$$ $$|f(x) - f(x_0$)| \leq C|x - x_0$|$$, what can you say about the sequence $$(u_n)$$?

Accepted Answer

$$(u_n)$$ converges to a fixed point of $$f$$

Question 14

Let $$J_f(x) = \$$int_0$$^1 f(t) \sin(xt) dt. $$What is the parity of $$J_f(x)$$?

Accepted Answer

Odd

Question 15

For $$F(x) = \frac{\sin x}{x}$$, what is $$\lim_{x 	o +\infty} F(x)$$?

Accepted Answer

$$0$$

Real Analysis Assignment: Limits and Differentiability of a Cubic Function

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