Question 1

Evaluate $$\lim\limits_{x 	o 0} \frac{\sin(4x)}{x}.$$

Accepted Answer

$$4$$

Question 2

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

Accepted Answer

$$2x e^x + x^2$ e^x - 10x + \frac{1}{x}$$

Question 3

Using the first principles, find $$f'(x)$$ for $$f(x) = x^2$ + 2x.$$

Accepted Answer

$$2x + 2$$

Question 4

Differentiate $$f(x) = \sin(2x + 1)$$ using the chain rule.

Accepted Answer

$$2 \cos(2x + 1)$$

Question 5

Find the equation of the tangent line to $$f(x) = x^2$ + 3x at$$ $$x = 2.$$

Accepted Answer

$$y = 7x - 6$$

Question 6

Evaluate $$\lim\limits_{x 	o 0} \frac{1 - \cos(2x)}{x^2$}.$$

Accepted Answer

$$4$$

Question 7

Differentiate $$f(x) = (2x + 3$$)^{x^2}$$ using logarithmic differentiation.

Accepted Answer

$$(2x + 3$$)^{x^2}$$ \left( \frac{2x^2$}{2x + 3} + 2x \ln(2x + 3) \right)$$

Question 8

Implicitly differentiate $$x^2$$ + $$y^2 = 25$ to find $$\frac{dy}{dx}$$.

Accepted Answer

$$\frac{-x}{y}$$

Question 9

A medication's concentration in the bloodstream is modeled by $$C(t) = e$$^{-0.5t}$$. Find the rate of change of concentration at t = 2$ hours.

Accepted Answer

$$-0.5 e$$^{-1}$$

Question 10

Find the derivative of $$f(x) = 	an(x).$$

Accepted Answer

$$\sec^2$(x)$$

Question 11

Evaluate $$\lim\limits_{x 	o 0} \frac{	an(5x)}{x}.$$

Accepted Answer

$$5$$

Question 12

Differentiate $$f(x) = \ln(x^2$ + 1).$$

Accepted Answer

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

Question 13

Find the equation of the tangent line to $$f(x) = \sin(x) at$$ $$x = 0.$$

Accepted Answer

$$y = x$$

Question 14

Implicitly differentiate $$x^2 + 2xy$$ + $$y^2 = 5$ to find $$\frac{dy}{dx}$$.

Accepted Answer

$$\frac{-x - y}{x + y}$$

Question 15

Differentiate $$f(x) = e$$^{x^2}$$ using the chain rule.

Accepted Answer

$$2x e$$^{x^2}$$

Question 16

A population grows according to $$P(t) = 100e$$^{0.03t}$$. What is the instantaneous rate of growth at t = 10$?

Accepted Answer

$$3e^{0.3}$$

Comprehensive Study Guidance for Calculus I Exam 2

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