Q: If $$y = \sec(x)$$, what is $$\frac{d^2$y}{dx^2$}$$?

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

Q: Given $$y = (\tan x + 10$$)^{21}$$, what is $$\frac{dy}{dx}$$?

$$21(\tan x + 10$$)^{20}$$ \sec^2$ x$$

Question 1

Suppose the position of a particle moving along a line is given by $$s(t) = t^3$ - 6t^2$ + 9t + 2. $$What is the acceleration of the particle at $$t = 2$$?

Accepted Answer

$$a(2) = -6$$

Question 2

Given $$y = x^2$ e$$^{3x}$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$2x e$$^{3x}$$ + 3x^2$ e$$^{3x}$$

Question 3

A stone is thrown vertically upward from a bridge with position function $$s(t) = -16t^2$ + 64t + 96. At $$what time does the stone reach its maximum height?

Accepted Answer

$$t = 2$$

Question 4

If $$y = \ln(x^2$ + 1)$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

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

Question 5

Find the derivative of $$f(x) = \$$arcsin(x^2$$ - 1).$$

Accepted Answer

$$\frac{2x}{\sqrt{1 - (x^2$ - 1)^2$}}$$

Question 6

Given $$y = 	an$$^{-1}$$(3x)$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$\frac{3}{1 + 9x^2$}$$

Question 7

If $$y = \ln(4x)$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$\frac{1}{x}$$

Question 8

A circular oil spill is expanding so that its radius increases at a rate of $$30 m/hr. $$How fast is the area increasing when the radius is $$100 m$$?

Accepted Answer

$$6000\pi m$^2/hr$$

Question 9

Given $$y = \frac{\ln(x^2$)}{x^2$}$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$\frac{2(1 - \ln(x^2$))}{x^3$}$$

Question 10

If $$y = \sec(x)$$, what is $$\frac{d^2$y}{dx^2$}$$?

Accepted Answer

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

Question 11

A plane is flying west at 120 mi/hr and another is flying north at 150 mi/hr. At a certain instant, the westbound plane is 180 mi from the airport and the northbound plane is 225 mi from the airport. How fast is the distance between the planes changing at that instant?

Accepted Answer

$$192 mi/hr ($$decreasing)

Question 12

Find the equation of the tangent line to $$f(x) = \frac{x^2$ + 1}{x^2$ - 4} at $$the point $$(3, 2).$$

Accepted Answer

$$y = \frac{6}{5}x + \frac{28}{5}$$

Question 13

If $$y = \sin$$^{-1}$$(x^2 - 1$$)$$, what is $$\frac{dy}{dx}$$ for -1$$ \u003c x \u003c 1$?

Accepted Answer

$$\frac{2x}{\sqrt{1 - (x^2$ - 1)^2$}}$$

Question 14

Given $$y = (	an x + 10$$)^{21}$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$21(	an x + 10$$)^{20}$$ \sec^2$ x$$

Question 15

If $$y = x \ln x$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$\ln x + 1$$

Question 16

Find the third derivative of $$f(x) = 3x^3$ - 5x + 12.$$

Accepted Answer

$$18$$

Question 17

Which of the following is the correct derivative of $$y = e$$^{-3x}$$?

Accepted Answer

$$-3e^{-3x}$$

Question 18

If $$y = \csc x$$, what is $$\frac{d^2$y}{dx^2$}$$?

Accepted Answer

$$\csc x (\cot^2$ x + \csc^2$ x)$$

Comprehensive Study Guide: Derivatives and Applications (MAC2311 Exam 2)

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