Question 1

Evaluate the following limit: $$\lim\limits_{x 	o -2} \frac{x^2$ - 4}{x + 2}$$

Accepted Answer

$$-4$$

Question 2

Given the piecewise function $$f(x) = \begin{cases} x + 4 & x \u003c 1 \ 2x - 1 & 1 \leq x \u003c 4 \ x^2$ & x \geq 4 \end{cases}$$, evaluate $$\lim\limits_{x 	o 1^-} f(x).$$

Accepted Answer

$$5$$

Question 3

Apply the Squeeze (Sandwich) Theorem to evaluate $$\lim\limits_{x 	o 0} x^2$ \cos\left(\frac{1}{x}ight).$$

Accepted Answer

$$0$$

Question 4

Find all values of $$a$$ that make the function $$f(x) = \begin{cases} x^2$ + a & x \u003c 2 \ ax + 1 & x \geq 2 \end{cases}$$ continuous.

Accepted Answer

$$a = 3$$

Question 5

Let $$f(x) = x^2$. Find f$$'(x)$$ using the limit definition of the derivative.

Accepted Answer

$$2x$$

Question 6

Find the derivative of $$f(x) = x^3$ - 2x^2$ + x - 5.$$

Accepted Answer

$$3x^2 - 4x + 1$

Question 7

For $$f(x) = x^4$ - 4x^3$ + 6x^2$, determine where f(x$$)$$ is increasing.

Accepted Answer

$$x  3$$

Question 8

A tank is being filled so that the volume $$V (in $$gallons) at time $$t (in $$minutes) is given by $$V(t) = t^2$ + 2t. At$$ $$t = 10$$ minutes, it contains 120 gallons. Estimate the rate at which water is flowing into the tank at $$t = 10$$ minutes.

Accepted Answer

$$22$$ gallons/min$

Question 9

Find the global maximum and minimum of $$f(x) = x^3$ - 3x^2$ + 1 on$$ $$[0, 3].$$

Accepted Answer

Global max at $$x = 3$$, min at $$x = 2$$

Question 10

Given $$f(x) = \sin(x)$$, find all points of inflection on $$[0, 2\pi].$$

Accepted Answer

$$x = 0, \pi$$

Question 11

Evaluate $$\lim\limits_{x 	o 0} \frac{\sin(3x)}{x}.$$

Accepted Answer

$$3$$

Question 12

If $$f(x) = |x - 2|$$, at which point is $$f(x)$$ not differentiable?

Accepted Answer

$$x = 2$$

Question 13

Suppose $$f(x) is $$continuous on $$[a, b]$$ and differentiable on $$(a, b). $$Which theorem guarantees there is a $$c in$$ $$(a, b)$$ such that $$f'(c) = \frac{f(b) - f(a)}{b - a}$$?

Accepted Answer

Mean Value Theorem

Question 14

Find the derivative of $$f(x) = \frac{1}{x}.$$

Accepted Answer

$$-\frac{1}{x^2$}$$

Question 15

If $$f(x) = x^2$ - 4x + 3$$, for which $$x is$$ $$f(x)$$ decreasing?

Accepted Answer

$$x \u003c 2$$

Step-by-Step Guidance for Calculus Final Exam Review

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