Q: Let $$y = \dfrac{x}{\sqrt{\tan x}}. $$Find $$\dfrac{dy}{dx}.$$

$$\dfrac{\sqrt{\tan x} - x \sec$$^{2}$$x / (2\sqrt{\tan x})}{\tan x}$$

Q: If $$f(x) = \sqrt{\tan x}$$, find $$f'(x).$$

$$\dfrac{1}{2\sqrt{\tan x}} \cdot \sec$$^{2}$$x$$

Question 1

Given $$y = 2\sin(x) - e$$^{x}$$ - 4$$, what is $$\dfrac{dy}{dx}$$?

Accepted Answer

$$2\cos(x) - e$$^{x}$$

Question 2

Let $$h(x) = 7 - 3x$$^{2}$$. What is h$$'(x)$$?

Accepted Answer

$$-6x$$

Question 3

Find the equation of the tangent line to the curve $$y = 7 - 3x$$^{2}$$ at the point (2$$, -5)$$.

Accepted Answer

$$y = -12x + 19$$

Question 4

Find the vertical asymptotes of $$f(x) = \dfrac{x$$^{2}$$ - 3x + 2}{x$$^{2}$$ - 4}.$$

Accepted Answer

$$x = 2, x = -2$$

Question 5

Evaluate $$\lim_{x 	o 2} \dfrac{3x - 2}{x - 2}.$$

Accepted Answer

$$\infty$$

Question 6

Given $$f(x) = x - 2$$, $$F(x) = x$$^{2}$$, and g(x$$) = \dfrac{1}{x}$$, find g(F(f(3$$)))$$.

Accepted Answer

$$\dfrac{1}{1}$$

Question 7

Determine $$\lim_{x 	o 2} \dfrac{7x - 12}{x - 2}. If $$the limit does not exist, enter DNE.

Accepted Answer

$$\infty$$

Question 8

If $$g(x) = 2$$, $$\lim_{x 	o a} f(x) = 2$$, and $$\lim_{x 	o a} g(x) = 2$$, is $$f(x)$$ continuous at $$x = a$$?

Accepted Answer

Yes

Question 9

If $$f(x) = \left(2x - \dfrac{1}{x}\$$right)^{2}$$, find f$$'(x)$$.

Accepted Answer

$$4x - 2x$$^{-2}$$

Question 10

Evaluate $$\lim_{x 	o 3} \dfrac{x$$^{2}$$ - 9}{x - 3}. If $$the limit does not exist, enter DNE.

Accepted Answer

$$6$$

Question 11

Let $$f(x) = x$$^{2}$$. Find the equation of the tangent line to the graph at x = -3$.

Accepted Answer

$$y = -6x + 9$$

Question 12

Determine the derivative of $$f(x) = \dfrac{x$$^{2}$$ + 2x}{x$$^{2}$$ + x}.$$

Accepted Answer

$$\dfrac{(2x + 2)(x$$^{2}$$ + x) - (x$$^{2}$$ + 2x)(2x + 1)}{(x$$^{2}$$ + x$$)^{2}$$}$$

Question 13

Sketch the graph of $$f(x) = \dfrac{x$$^{2}$$ - 3x + 2}{x$$^{2}$$ - 4}$$ and find $$\lim_{x 	o 2} f(x).$$

Accepted Answer

$$DNE$$

Question 14

Let $$y = \dfrac{x}{\sqrt{	an x}}. $$Find $$\dfrac{dy}{dx}.$$

Accepted Answer

$$\dfrac{\sqrt{	an x} - x \sec$$^{2}$$x / (2\sqrt{	an x})}{	an x}$$

Question 15

If $$f(x) = \sqrt{	an x}$$, find $$f'(x).$$

Accepted Answer

$$\dfrac{1}{2\sqrt{	an x}} \cdot \sec$$^{2}$$x$$

Question 16

The limit $$\lim_{h 	o 0} \dfrac{8 + h$$^{2}$$ - 216}{h} is $$the standard expression defining the derivative of some function $$f(x) at $$some number $$a. $$Find $$f(x)$$ and $$a.$$

Accepted Answer

$$f(x) = x$$^{2}$$, a = 8$$

Fundamental Concepts in Calculus: Derivatives, Limits, and Tangents

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