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 $$f(x) = 0$$ have 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}$$, what can be said about u_n$ for all n$$ \in \mathbb{N}$$?

Accepted Answer

$$0 \leq u_n \leq \beta$$

Question 7

Show that $$u_{n+1} - u_n = f(u_n)$$ for the sequence $$u_{n+1} = 1 + \frac{u_n^3$}{12}$$ and deduce the monotonicity of $$(u_n). $$What is the behavior?

Accepted Answer

$$(u_n) is $$decreasing and bounded below

Question 8

What is the limit of the sequence $$(u_n)$$ defined by $$u_0 = 1$, u$$_{n+1} = 1 + \frac{u_$$n^3$$}{12}$$?

Accepted Answer

$$\beta$$

Question 9

Let $$F(x) = \frac{\sin x}{x}$$ and $$G(x) = \frac{1 - \cos x}{x}$$ for $$x \u003e 0. $$Which of the following is true about their continuity at $$x = 0$$?

Accepted Answer

Both $$F$$ and $$G$$ can be extended by continuity at $$x = 0$$

Question 10

What is the value of $$\lim_{x 	o 0} \frac{\sin x}{x}$$?

Accepted Answer

$$1$$

Question 11

Let $$F(x) = \frac{\sin x}{x}. $$The positive real numbers $$a_k$$ such that $$F(a_k) = 0$$ form a strictly increasing sequence. What is $$a_k$$?

Accepted Answer

$$a_k = k\pi$$

Question 12

Let $$f be a $$real function such that $$|f(x) - f(x_0$)| \leq C|x - x_0$|^\alpha$$ for all $$x, x_0$ \in \mathbb{R}$$, with $$\alpha \u003e 1. $$What can be deduced about $$f$$?

Accepted Answer

$$f is $$constant

Question 13

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

Accepted Answer

$$\alpha = 1$$

Question 14

Let $$F(x) = \frac{\sin x}{x}$$ and $$G(x) = \frac{1 - \cos x}{x}. $$What is the relationship between the zeros of $$F$$ and $$G$$?

Accepted Answer

The zeros of $$G$$ are at $$b_k = 2k\pi$$

Question 15

Let $$f be a $$real function such that $$|f(x) - f(x_0$)| \leq C|x - x_0$|^\alpha$$ for $$\alpha \u003e 1. $$What is $$f'(x)$$ for all $$x \in \mathbb{R}$$?

Accepted Answer

$$f'(x) = 0$$

Question 16

Suppose $$f is $$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

$$I_f(x) is $$even

Question 17

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

Accepted Answer

$$J_f(x) is $$odd

Question 18

Let $$f be as $$above. What is $$\lim_{x 	o +\infty} I_f(x)$$?

Accepted Answer

$$0$$

Analyzing the Limit of a Cubic Function as x Approaches Infinity

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