Q: Given $R(x) = 800\sqrt{2x - 5}$ and $C(x) = 172\sqrt{x} + 200$, what is the rate of change of total profit $P(x)$ when $x$ items have been produced and sold?

$P'(x) = 800 \cdot \frac{1}{\sqrt{2x - 5}} \cdot 2 - 172 \cdot \frac{1}{2\sqrt{x}}$

Q: Find the derivative of $f(x) = x^2\sqrt{x} - 8\sqrt{x}$.

$f'(x) = 2x\sqrt{x} + x^2 \cdot \frac{1}{2\sqrt{x}} - 4x^{-1/2}$

Q: Find the derivative of $f(x) = \frac{x^2 + 4x + 2}{x - 3}$ using the quotient rule.

$f'(x) = \frac{(x - 3)(2x + 4) - (x^2 + 4x + 2)(1)}{(x - 3)^2}$

Question 1

Given the function $f(x) = \begin{cases} -3x + 7, & x < 1 \ 4x - 1, & x \geq 1 \end{cases}$, what is $\lim_{x 	o 1^-} f(x)$?

Accepted Answer

$-3(1) + 7 = 4$

Question 2

For the function $f(x) = x^2 - 9$ for $x < 5$ and $x^2 + c$ for $x \geq 5$, what value of $c$ makes $\lim_{x 	o 5} f(x)$ exist?

Accepted Answer

$c = 41$

Question 3

Suppose the total number of people infected with a virus after $t$ weeks is given by $P(t)$. If $P(0) = 220$ and as $t$ increases, $P$ approaches $1540$, what is $\lim_{t 	o \infty} P(t)$?

Accepted Answer

$1540$

Question 4

Find $\lim_{x 	o 2} \frac{x^2 + 4x - 28}{x - 2}$.

Accepted Answer

$8$

Question 5

Given $f(x) = x^2 - 4x$, what is the simplified form of the difference quotient $\frac{f(x+h) - f(x)}{h}$?

Accepted Answer

$2x + h - 4$

Question 6

A city’s population grows according to $P(t) = 250{,}000 + 3{,}000t$. What is the rate of change of the population at $t = 10$?

Accepted Answer

$60{,}000$

Question 7

Find the derivative of $y = x^2 - 3x^2 + 10x = 100$.

Accepted Answer

$y' = 2x - 6x + 10$

Question 8

Given $f(x) = 12x\sqrt{x} - 3x$, find $f'(x)$.

Accepted Answer

$f'(x) = 18\sqrt{x} - 3$

Question 9

For the function $f(x) = x^3 - x^2 - 3x + 1$, at which $x$ values is the tangent line horizontal?

Accepted Answer

$x = 3$ and $x = 1$

Question 10

Given $R(x) = 800\sqrt{2x - 5}$ and $C(x) = 172\sqrt{x} + 200$, what is the rate of change of total profit $P(x)$ when $x$ items have been produced and sold?

Accepted Answer

$P'(x) = 800 \cdot \frac{1}{\sqrt{2x - 5}} \cdot 2 - 172 \cdot \frac{1}{2\sqrt{x}}$

Question 11

Find the derivative of $y = (x^2 + 2)^5$.

Accepted Answer

$y' = 5(x^2 + 2)^4 \cdot 2x$

Question 12

Suppose the revenue from selling $x$ units is $R(x) = 0.01x^2 + 1000x$. What is the average rate of revenue per unit when $x$ changes from 300 to 303?

Accepted Answer

$994$

Question 13

Given $f(x) = x^2 - 4x$, what is the equation of the tangent line at $x = 2$?

Question 14

Find the derivative of $f(x) = x^2\sqrt{x} - 8\sqrt{x}$.

Accepted Answer

$f'(x) = 2x\sqrt{x} + x^2 \cdot \frac{1}{2\sqrt{x}} - 4x^{-1/2}$

Question 15

For the function $u(t) = 7.35 + 1.97t - 0.0076t^2 + 0.0000252t^3$, what is the rate of change of weight with respect to time at $t = 13$ months?

Accepted Answer

$u'(13) = 1.91$ lbs/month

Question 16

Given $T(t) = \frac{9t}{t^2 + 6t} + 98.6$, what is $T'(1)$, the rate of change of temperature at $t = 1$ hour?

Accepted Answer

$0.84$ degrees per hour

Question 17

Suppose the distance of a car at time $t$ is $s(t) = 10t^2$. What is the acceleration at $t = 4$ seconds?

Accepted Answer

$a(4) = 20$

Question 18

Find the derivative of $f(x) = \frac{x^2 + 4x + 2}{x - 3}$ using the quotient rule.

Accepted Answer

$f'(x) = \frac{(x - 3)(2x + 4) - (x^2 + 4x + 2)(1)}{(x - 3)^2}$

Business Calculus Study Guide: Limits, Derivatives, and Applications

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