Q: Given $y = x^2 e^x$, what is the equation of the tangent line at $x=1$?

$y - e = 3e(x - 1)$,\ \text{where}\ e = e^1$

Q: If $f(x) = (\tan(x^2 + 1))(3x^2 - 2x)^2$, what is $f'(x)$?

$f'(x) = \sec^2(x^2 + 1) \cdot 2x (3x^2 - 2x)^2 + \tan(x^2 + 1) \cdot 2(3x^2 - 2x)(6x - 2)$

Question 1

Given the function $f(x) = (x^3 - 8x + x)^4$, what is $f'(x)$?

Accepted Answer

$f'(x) = 4(x^3 - 8x + x)^3 (3x^2 - 8)$

Question 2

A cylinder has a radius $r = 3$ (decreasing at $1$ unit/sec) and height $h = 5$ (increasing at $2$ units/sec). What is the rate of change of its volume $V = \pi r^2 h$ at this instant?

Accepted Answer

$\frac{dV}{dt} = 12\pi$

Question 3

Find the equation of the tangent line to $x - y^2 = x^2 - 1$ at the point $(1, \sqrt{2})$.

Accepted Answer

$y - \sqrt{2} = -1(x - 1)$

Question 4

What are the maximum and minimum values of $f(x) = \sqrt{x^2 - x^3}$ on $[0,1]$?

Accepted Answer

Maximum at $x=\frac{2}{3}$, minimum at $x=0$

Question 5

A 6 ft tall person walks away from a 15 ft lamppost at 2 ft/sec. When the person is 4 ft from the lamppost, how fast is the tip of their shadow moving?

Accepted Answer

$3$ ft/sec

Question 6

Find the equation of the tangent line to $x + \log(xy) = x + y$ at $(1,1)$.

Accepted Answer

$y - 1 = -1(x - 1)$

Question 7

A kite is held at a constant altitude of 30 ft. The string is let out at 2 ft/sec when the horizontal distance is 40 ft. What is the rate of change of the angle of inclination $	heta$ (in radians/sec) at that instant?

Accepted Answer

$\frac{1}{52}$

Question 8

Find the maximum and minimum of $f(x) = (x^2 - 2x + 1) + \sqrt{x}$ on $[0,1]$.

Accepted Answer

Maximum at $x=1$, minimum at $x=0$

Question 9

Given $y = x^2 e^x$, what is the equation of the tangent line at $x=1$?

Accepted Answer

$y - e = 3e(x - 1)$,\ 	ext{where}\ e = e^1$

Question 10

A rectangular pen must enclose an area of 600. Three sides cost $10/unit$ and the fourth side costs $30/unit$. What is the shape of the pen that minimizes cost?

Accepted Answer

A rectangle with the expensive side as the longer side

Question 11

The height of a firework is $s(t) = -24t^2 + 96t$. What is the maximum height reached by the firework?

Accepted Answer

$192$ ft

Question 12

For $s(t) = -24t^2 + 96t$, at what time does the firework hit the ground?

Accepted Answer

$t = 0$ and $t = 4$

Question 13

Let $g(x) = -7x^2 + 32x$. Find the equation of the tangent line at $x = -1$.

Accepted Answer

$y + 39 = 46(x + 1)$

Question 14

Let $f(x) = -2x + 2 + 2^x$. What are the zeros of $f(x)$?

Accepted Answer

$x = $ solution to $-2x + 2 + 2^x = 0$

Question 15

Let $f(x) = -2x + 2 + 2^x$. What is the $x$-coordinate of the critical point?

Accepted Answer

$x = \log_2(2)$

Question 16

Find the derivative of $f(x) = \sin(\sin x)$.

Accepted Answer

$f'(x) = \cos(\sin x) \cdot \cos x$

Question 17

Find the derivative of $f(x) = \log(e^{x} 	an x)$.

Accepted Answer

$f'(x) = 1 + \sec^2 x$

Question 18

Find the derivative of $f(x) = \log(\log x)$.

Accepted Answer

$f'(x) = \frac{1}{x \log x}$

Question 19

If $f(x) = (	an(x^2 + 1))(3x^2 - 2x)^2$, what is $f'(x)$?

Accepted Answer

$f'(x) = \sec^2(x^2 + 1) \cdot 2x (3x^2 - 2x)^2 + 	an(x^2 + 1) \cdot 2(3x^2 - 2x)(6x - 2)$

Calculus Practice Assessment 2: Derivatives, Related Rates, Optimization, and Applications

Study Guide - Practice Questions

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