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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