What must be true of $F(x)$ and $G(x)$ if both are antiderivatives of $f(x) ?$

Amy J.

Numerade Educator

Explain why the restriction $n \neq-1$ is necessary in the rule

$$

\int x^{n} d x=\frac{x^{n+1}}{n+1}+C

$$

Amy J.

Numerade Educator

Find the following.

$$

\int\left(4 z^{3}+3 z^{2}+2 z-6\right) d z

$$

Amy J.

Numerade Educator

Find the following.

$$

\int\left(16 y^{3}+9 y^{2}-6 y+3\right) d y

$$

Amy J.

Numerade Educator

Find the following.

$$

\int\left(t^{1 / 4}+\pi^{1 / 4}\right) d t

$$

Amy J.

Numerade Educator

Find the following.

$$

\int\left(4 \sqrt{v}-3 v^{3 / 2}\right) d v

$$

Amy J.

Numerade Educator

Find the following.

$$

\int\left(10 u^{3 / 2}-14 u^{5 / 2}\right) d u

$$

Amy J.

Numerade Educator

Find the following.

$$

\int\left(56 t^{5 / 2}+18 t^{7 / 2}\right) d t

$$

Amy J.

Numerade Educator

Find the following.

$$

\int\left(\frac{\pi^{3}}{y^{3}}-\frac{\sqrt{\pi}}{\sqrt{y}}\right) d y

$$

Amy J.

Numerade Educator

Find the following.

$$

\int\left(\sqrt{u}+\frac{1}{u^{2}}\right) d u

$$

Amy J.

Numerade Educator

Find the following.

$$

\int\left(-9 t^{-2.5}-2 t^{-1}\right) d t

$$

Amy J.

Numerade Educator

Find the following.

$$

\int\left(\frac{-3}{x}+4 e^{-0.4 x}+e^{0.1}\right) d x

$$

Amy J.

Numerade Educator

Find the following.

$$

\int\left(\frac{9}{x}-3 e^{-0.4 x}\right) d x

$$

Amy J.

Numerade Educator

Find the following.

$$

\int \frac{1-2 \sqrt[3]{z}}{\sqrt[3]{z}} d z

$$

Amy J.

Numerade Educator

Find an equation of the curve whose tangent line has a slope of

$$f^{\prime}(x)=x^{2 / 3}$$

given that the point $(1,3 / 5)$ is on the curve.

Amy J.

Numerade Educator

The slope of the tangent line to a curve is given by

$$f^{\prime}(x)=6 x^{2}-4 x+3$$

If the point $(0,1)$ is on the curve, find an equation of the curve.

Amy J.

Numerade Educator

Cost Find the cost function for each marginal cost function.

$$

C^{\prime}(x)=4 x-5 ; \text { fixed cost is } \$ 8

$$

Amy J.

Numerade Educator

Cost Find the cost function for each marginal cost function.

$$

C^{\prime}(x)=0.2 x^{2}+5 x ; \quad \text { fixed cost is } \$ 10

$$

Amy J.

Numerade Educator

Cost Find the cost function for each marginal cost function.

$$

C^{\prime}(x)=0.03 e^{0.01 x} ; \quad \text { fixed cost is } \$ 8

$$

Amy J.

Numerade Educator

Cost Find the cost function for each marginal cost function.

$$

C^{\prime}(x)=x^{1 / 2} ; \quad 16 \text { units } \operatorname{cost} \$ 45

$$

Amy J.

Numerade Educator

Cost Find the cost function for each marginal cost function.

$$

C^{\prime}(x)=x^{2 / 3}+2 ; \quad 8 \text { units cost } \$ 58

$$

Amy J.

Numerade Educator

Cost Find the cost function for each marginal cost function.

$$

C^{\prime}(x)=x+1 / x^{2} ; \quad 2 \text { units cost } \$ 5.50

$$

Amy J.

Numerade Educator

Cost Find the cost function for each marginal cost function.

$$

C^{\prime}(x)=5 x-1 / x ; \quad 10 \text { units } \operatorname{cost} \$ 94.20

$$

Amy J.

Numerade Educator

Cost Find the cost function for each marginal cost function.

$$

C^{\prime}(x)=1.2^{x}(\ln 1.2) ; \quad 2 \text { units cost } \$ 9.44

$$

Amy J.

Numerade Educator

Demand Find the demand function for each marginal revenue function. Recall that if no items are sold, the revenue is $0 .$

$$

R^{\prime}(x)=175-0.02 x-0.03 x^{2}

$$

Amy J.

Numerade Educator

Demand Find the demand function for each marginal revenue function. Recall that if no items are sold, the revenue is $0 .$

$$

R^{\prime}(x)=50-5 x^{2 / 3}

$$

Amy J.

Numerade Educator

Demand Find the demand function for each marginal revenue function. Recall that if no items are sold, the revenue is $0 .$

$$

R^{\prime}(x)=500-0.15 \sqrt{x}

$$

Amy J.

Numerade Educator

Demand Find the demand function for each marginal revenue function. Recall that if no items are sold, the revenue is $0 .$

$$

R^{\prime}(x)=600-5 e^{0.0002 x}

$$

Amy J.

Numerade Educator

Text Messaging The approximate rate of change in the number (in billions) of monthly text messages is given by

$$f^{\prime}(t)=7.50 t-16.8$$

where $t$ represents the number of years since 2000 . In 2005 $(t=5)$ there were approximately 9.8 billion monthly text messages. Source: Cellular Telecommunication \& Internet Association.

a. Find the function that gives the total number (in billions) of monthly text messages in year $t .$

b. According to this function, how many monthly text messages were there in 2009$?$ Compare this with the actual number of 152.7 billion.

Amy J.

Numerade Educator

Profit The marginal profit of a small fast-food stand is given, in thousands of dollars, by

$$P^{\prime}(x)=\sqrt{x}+\frac{1}{2}$$

where $x$ is the sales volume in thousands of hamburgers. The profit is $-\$ 1000$ when no hamburgers are sold. Find the profit function.

Amy J.

Numerade Educator

Profit The marginal profit in dollars on Brie cheese sold at a cheese store is given by

$$P^{\prime}(x)=x\left(50 x^{2}+30 x\right)$$

where $x$ is the amount of cheese sold, in hundreds of pounds. The profit is $-\$ 40$ when no cheese is sold.

a. Find the profit function.

b. Find the profit from selling 200 lb of Brie cheese.

Amy J.

Numerade Educator

Biochemical Excretion If the rate of excretion of a bio-chemical compound is given by

$$f^{\prime}(t)=0.01 e^{-0.01 t}$$

the total amount excreted by time $t$ (in minutes) is $f(t)$

a. Find an expression for $f(t)$

b. If 0 units are excreted at time $t=0,$ how many units are excreted in 10 minutes?

Amy J.

Numerade Educator

Flour Beetles A model for describing the population of adult flour beetles involves evaluating the integral

$$\int \frac{g(x)}{x} d x$$

where $g(x)$ is the per-unit-abundance growth rate for a population of size $x$ . The researchers consider the simple case in which $g(x)=a-b x$ for positive constants $a$ and $b$ . Find the integral in this case. Source: Ecology.

Amy J.

Numerade Educator

Concentration of a Solute According to Fick's law, the diffusion of a solute across a cell membrane is given by

$$c^{\prime}(t)=\frac{k A}{V}[C-c(t)]$$

where $A$ is the area of the cell membrane, $V$ is the volume of the cell, $c(t)$ is the concentration inside the cell at time $t, C$ is the concentration outside the cell, and $k$ is a constant. If $c_{0}$ represents the concentration of the solute inside the cell when $t=0,$ then it can be shown that

$$c(t)=\left(c_{0}-C\right) e^{-k t t / V}+C$$

a. Use the last result to find $c^{\prime}(t)$ .

b. Substitute back into Equation $(1)$ to show that $(2)$ is indeed the correct antiderivative of $(1)$ .

Sajin S.

Numerade Educator

Cell Growth Under certain conditions, the number of cancer cells $N(t)$ at time $t$ increases at a rate

$$N^{\prime}(t)=A e^{k t}$$

where $A$ is the rate of increase at time 0 (in cells per day) and $k$ is a constant.

a. Suppose $A=50$ , and at 5 days, the cells are growing at a rate of 250 per day. Find a formula for the number of cells after $t$ days, given that 300 cells are present at $t=0$ .

b. Use your answer from part a to find the number of cells present after 12 days.

Check back soon!

Blood Pressure The rate of change of the volume $V(t)$ of blood in the aorta at time $t$ is given by

$$V^{\prime}(t)=-k P(t)$$

where $P(t)$ is the pressure in the aorta at time $t$ and $k$ is a constant that depends upon properties of the aorta. The pressure in the aorta is given by

$$P(t)=P_{0} e^{-m t}$$

where $P_{0}$ is the pressure at time $t=0$ and $m$ is another constant. Letting $V_{0}$ be the volume at time $t=0$ , find a formula for $V(t) .$

Sajin S.

Numerade Educator

Bachelor's Degrees The number of bachelor's degrees conferred in the United States has been increasing steadily in recent decades. Based on data from the National Center for Education Statistics, the rate of change of the number of bachelor's degrees (in thousands) can be approximated by the function

$$B^{\prime}(t)=0.06048 t^{2}-1.292 t+15.86$$

where $t$ is the number of years since $1970 .$ Source: National Center for Education Statistics.

a. Find $B(t),$ given that about $839,700$ degrees were conferred in 1970$(t=0)$ .

b. Use the formula from part a to project the number of bachelor's degrees that will be conferred in 2015$(t=45)$ .

Sajin S.

Numerade Educator

Degrees in Dentistry The number of degrees in dentistry (D.D.S. or D.M.D.) conferred to females in the United States has been increasing steadily in recent decades. Based on data from the National Center for Education Statistics, the rate of change of the number of bachelor's degrees can be approximated by the

function

$$D^{\prime}(t)=29.25 e^{0.03572 t}$$

where $t$ is the number of years since $1980 .$ Source: National Center for Education Statistics.

a. Find $D(t),$ given that about 700 degrees in dentistry were conferred to females in 1980$(t=0)$

b. Use the formula from part a to project the number of degrees in dentistry that will be conferred to females in 2015$(t=35)$ .

Check back soon!

Exercises $67-71$ refer to Example 11 in this section.

Velocity For a particular object, $a(t)=5 t^{2}+4$ and $v(0)=6 .$ Find $v(t) .$

Amy J.

Numerade Educator

Exercises $67-71$ refer to Example 11 in this section.

Distance Suppose $v(t)=9 t^{2}-3 \sqrt{t}$ and $s(1)=8 .$ Find $s(t) .$

Amy J.

Numerade Educator

Exercises $67-71$ refer to Example 11 in this section.

Time An object is dropped from a small plane flying at 6400 ft. Assume that $a(t)=-32$ ft per second and $v(0)=0$ . Find $s(t) .$ How long will it take the object to hit the ground?

Amy J.

Numerade Educator

Exercises $67-71$ refer to Example 11 in this section.

Distance Suppose $a(t)=18 t+8, v(1)=15, $ and $s(1)=19 .$ Find $s(t)$

Amy J.

Numerade Educator

Exercises $67-71$ refer to Example 11 in this section.

Distance Suppose $a(t)=(15 / 2) \sqrt{t}+3 e^{-t}, v(0)=-3,$ and $s(0)=4 .$ Find $s(t) .$

Amy J.

Numerade Educator

Motion Under Gravity Show that an object thrown from an initial height $h_{0}$ with an initial velocity $v_{0}$ has a height at time $t$ given by the function

$$h(t)=\frac{1}{2} g t^{2}+v_{0} t+h_{0}$$

where $g$ is the acceleration due to gravity, a constant with value $-32 \mathrm{ft} / \mathrm{sec}^{2}$

Check back soon!

Rocket A small rocket was launched straight up from a plat-form. After 5 seconds, the rocket reached a maximum height of 412 $\mathrm{ft}$ . Find the initial velocity and height of the rocket. Hint: See the previous exercise.)

Sajin S.

Numerade Educator

Rocket Science In the 1999 movie October Sky, Homer Hickum was accused of launching a rocket that started a forest fire. Homer proved his innocence by showing that his rocket could not have flown far enough to reach where the fire started. He used the following reasoning.

a. Using the fact that $a(t)=-32$ (see Example 11$(b) )$ , find $v(t)$ and $s(t),$ given $v(0)=v_{0}$ and $s(0)=0$ . The initial velocity was unknown, and the initial height was 0 $\mathrm{ft.}$ )

b. Homer estimated that the rocket was in the air for 14 seconds. Use $s(14)=0$ to find $v_{0}$ .

c. If the rocket left the ground at a $45^{\circ}$ angle, the velocity in the horizontal direction would be equal to $v_{0},$ the velocity in the vertical direction, so the distance traveled horizontally so this would overestimate the ground at a steeper angle, landing point.) Find the distance the rocket would travel horizontally during its 14-second flight.

Sajin S.

Numerade Educator