Resources

Resources

Bytelearn - cat image with glasses

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

How does $f(t) = 10^t$ change over the interval from $t = 9$ to $t = 10$ ? $\newline$ Choices: $\newline$ (A) $f(t)$ increases by $1,000\%$ $\newline$ (B) $f(t)$ decreases by a factor of $10$ $\newline$ (C) $f(t)$ increases by $10\%$ $\newline$ (D) $f(t)$ increases by $t = 9$ $0$

View step-by-step help

View step-by-step help

Find instantaneous rates of change

Full solution

Q. How does

f(t) = 10^t

change over the interval from

t = 9

to

t = 10

?

\newline

Choices:

\newline

(A)

f(t)

increases by

1,000\%

\newline

(B)

f(t)

decreases by a factor of

10

\newline

(C)

f(t)

increases by

10\%

\newline

(D)

f(t)

increases by

t = 9

0

Calculate $f(t)$ : Calculate $f(t)$ when $t = 9$ : $f(9) = 10^9$ .
Calculate $f(t)$ : Calculate $f(t)$ when $t = 10$ : $f(10) = 10^{10}$ .
Find increase: Find the increase from $f(9)$ to $f(10)$ : $f(10) - f(9) = 10^{10} - 10^{9}$ .
Factor out base: Factor out the common base to simplify: $10^9(10 - 1) = 10^9 \times 9$ .
Calculate expression: Calculate the simplified expression: $10^9 \times 9 = 9 \times 10^9$ .
Compare increase: Compare the increase to the original value $f(9)$ : $\frac{9 \times 10^9}{10^9} = 9$ .

More problems from Find instantaneous rates of change

Question

Consider the curve given by the equation

x y^{2}+5 x y=50

. It can be shown that

\frac{d y}{d x}=\frac{-y(y+5)}{x(2 y+5)} \text {. }

\newline

Write the equation of the vertical line that is tangent to the curve.

Posted 1 month ago

Question

The rate of change

\frac{d P}{d t}

of the number of algae in a tank is modeled by the following differential equation:

\newline

\frac{d P}{d t}=\frac{2317}{10614} P\left(1-\frac{P}{662}\right)

\newline

t=0

, the number of algae in the tank is

174

and is increasing at a rate of

28

algae per minute. At what value of

P

P(t)

growing the fastest?

\newline

Posted 2 months ago

Question

The rate of change

\frac{d P}{d t}

of the number of bacteria in a tank is modeled by the following differential equation:

\newline

\frac{d P}{d t}=\frac{2}{9849} P(598-P)

\newline

t=0

, the number of bacteria in the tank is

196

and is increasing at a rate of

16

bacteria per minute. At what value of

P

does the graph of

P(t)

have an inflection point?

\newline

Posted 2 months ago

Question

The rate of change

\frac{d P}{d t}

of the number of people infected by a disease is modeled by the following differential equation:

\newline

\frac{d P}{d t}=\frac{45}{125404} P(800-P)

\newline

t=0

, the number of people infected by the disease is

214

and is increasing at a rate of

45

people per hour. What is the limiting value for the total number of people infected by the disease as time increases?

\newline

Posted 2 months ago

Question

Which of the following is a correct interpretation of the expression

\newline

-4+13

\newline

1

\newline

(A) The number that is

4

-13

on the number line

\newline

(B) The number that is

4

to the right of

-13

on the number line

\newline

(C) The number that is

13

-4

on the number line

\newline

(D) The number that is

13

to the right of

-4

on the number line

Posted 2 months ago

Question

f

f(x)=x^{3}+2 \sin (3 x-3)

. Use a calculator to write the equation of the line tangent to the graph of

f

x=0.5

. You should round all decimals to

3

\newline

Posted 2 months ago

Question

f

f(x)=x^{2}-2 x+3 \cos \left(x^{2}-x\right)

. Use a calculator to write the equation of the line tangent to the graph of

f

x=-2.5

. You should round all decimals to

3

\newline

Posted 2 months ago

Question

f

f(x)=x^{2}+x-2 \sin (2 x)

. Use a calculator to write the equation of the line tangent to the graph of

f

x=3

. You should round all decimals to

3

\newline

Posted 2 months ago

Question

f

f(x)=x^{3}+\cos \left(2 x^{2}+5\right)

. Use a calculator to write the equation of the line tangent to the graph of

f

x=1

. You should round all decimals to

3

\newline

Posted 2 months ago

Question

f

f(x)=x^{3}+3-5 \sin \left(x^{2}\right)

. Use a calculator to write the equation of the line tangent to the graph of

f

x=1

. You should round all decimals to

3

\newline

Posted 2 months ago