$$\frac{\sec(x)\tan(x)}{2\sqrt{\sec(x)}}$$

Question 1

Find the derivative of the function $$f(x) = (1 + 2x^3$)\left( \frac{1}{3\sqrt{x}} + x \right). $$Which of the following is correct?

Accepted Answer

$$6x^2$$\left( \frac{1}{3\sqrt{x}} + x \right) + (1 + $$2x^3$$)\left( -\frac{1}{$$6x^{3/2}$$} + 1 \right)$$

Question 2

What is the equation of the tangent line to $$f(x) = x \sin(2x) at$$ $$x = \frac{\pi}{4}$$?

Accepted Answer

$$y = \frac{\pi}{4} + (1 + \frac{\pi}{2} \cdot 0)\left(x - \frac{\pi}{4}\right)$$

Question 3

Given $$f(2) = 5$$ and $$f'(2) = 3$$, let $$g(x) = 3x f(2x). $$What is the equation of the tangent line to $$g(x) at$$ $$x = 1$$?

Accepted Answer

$$y = 15 + 21(x-1)$$

Question 4

Using the table, what is $$\frac{d}{dx}[-3f(x)] at$$ $$x=3$$?

Accepted Answer

$$3$$

Question 5

Using the table, what is $$\frac{d}{dx}[x^2$ f(x)] at$$ $$x=2$$?

Accepted Answer

$$2 \cdot 2 \cdot 3 + 4 \cdot 9 = 12 + 36 = 48$$

Question 6

Find the derivative of $$f(x) = \sqrt{\sec(x)}.$$

Accepted Answer

$$\frac{\sec(x)	an(x)}{2\sqrt{\sec(x)}}$$

Question 7

Find the derivative of $$f(x) = 	an^2$(x).$$

Accepted Answer

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

Question 8

Implicitly differentiate $$e^x + 2y = 3xy$$ with respect to $$x.$$

Accepted Answer

$$e^x + 2\frac{dy}{dx} = 3y + 3x\frac{dy}{dx}$$

Question 9

For the implicitly defined curve $$x^3$$ + $$5y^2 = 7$, what is $$\frac{$$d^2y$$}{$$dx^2$$}$$ in terms of x$ and y$?

Accepted Answer

$$\frac{-3x^2$}{10y} - \frac{3x^2$}{10y^2$}\left(\frac{dy}{dx}\right)$$

Question 10

Suppose the position of a football thrown upwards is $$s(t) = -16t^2$ + 10t + 6. $$What is the velocity function $$v(t)$$?

Accepted Answer

$$v(t) = -32t + 10$$

Question 11

A 15-foot ladder leans against a wall. The bottom slides away at 2 ft/s. How fast is the top sliding down when the bottom is 9 ft from the wall?

Accepted Answer

$$-1.2 ft/s$$

Question 12

Air is pumped into a spherical balloon at $$100\ \mathrm{cm}^3/\mathrm{s}. $$How fast is the diameter increasing when the radius is 5 cm?

Accepted Answer

$$\frac{2}{5\pi} cm/s$$

Question 13

Find the derivative of $$y = (\sin(x$$))^{\sec(x)}$$.

Accepted Answer

$$(\sin(x$$))^{\sec(x)}$$\left( \sec(x)\cot(x) + \ln(\sin(x))\sec(x)	an(x) ight)$$

Question 14

Find the derivative of $$f(x) = \ln(x^3$ + 2).$$

Accepted Answer

$$\frac{3x^2$}{x^3$ + 2}$$

Question 15

Find the derivative of $$f(x) = 2\cos(3x).$$

Accepted Answer

$$-6\sin(3x)$$

Question 16

Find the derivative of $$f(x) = \cos(\sin(x^2$ - x + 9)).$$

Accepted Answer

$$-\sin(\sin(x^2$ - x + 9)) \cdot (2x - 1)$$

Exam 2 Review for Calculus I – Step-by-Step Guidance

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