Resources

Resources

Bytelearn - cat image with glasses

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

An object travels along a straight line. The function $v(t) = 4t^4 - 3t^3 - 11t + 3$ gives the object's velocity, in feet per hour, at time $t$ hours. $\newline$ Write a function that gives the object's acceleration $a(t)$ in feet per hour per hour. $\newline$ a(t) = ______

View step-by-step help

View step-by-step help

Relate position, velocity, speed, and acceleration using derivatives

Full solution

Q. An object travels along a straight line. The function

v(t) = 4t^4 - 3t^3 - 11t + 3

gives the object's velocity, in feet per hour, at time

t

hours.

\newline

Write a function that gives the object's acceleration

a(t)

in feet per hour per hour.

\newline

a(t) = ______

Identify Velocity Function: Identify the velocity function and the need to differentiate it to find acceleration. $\newline$ The velocity function $v(t) = 4t^4 - 3t^3 - 11t + 3$ needs to be differentiated to find the acceleration function $a(t)$ . $\newline$ Differentiate each term of $v(t)$ with respect to $t$ .
Differentiate $4t^4$ : Differentiate the first term $4t^4$ . Using the power rule, the derivative of $4t^4$ is $16t^3$ .
Differentiate $-3t^3$ : Differentiate the second term $-3t^3$ . Using the power rule, the derivative of $-3t^3$ is $-9t^2$ .
Differentiate $-11t$ : Differentiate the third term $-11t$ . Using the power rule, the derivative of $-11t$ is $-11$ .
Differentiate Constant Term: Differentiate the constant term $+3$ . The derivative of a constant is $0$ .
Combine Differentiated Terms: Combine all the differentiated terms to form the acceleration function. $\newline$ $a(t) = 16t^3 - 9t^2 - 11 + 0$ $\newline$ Simplify to $a(t) = 16t^3 - 9t^2 - 11$ .

More problems from Relate position, velocity, speed, and acceleration using derivatives

Question

A particle travels along the

x

-axis such that its velocity is given by

v(t)=t^{0.7} \sin (2 t+3)

. What is the acceleration of the particle at time

t=4

? You may use a calculator and round your answer to the nearest thousandth.

\newline

Posted 2 months ago

Question

A particle travels along the

x

-axis such that its velocity is given by

v(t)=t^{1.1} \cos \left(t^{2}-5\right)

. If the position of the particle is

x=-2

t=2.5

, what is the position of the particle when

t=1

? You may use a calculator and round your answer to the nearest thousandth.

\newline

Posted 2 months ago

Question

A particle travels along the

x

-axis such that its velocity is given by

v(t)=t^{0.9} \sin (t-1)

. If the position of the particle is

x=-3

t=3

, what is the position of the particle when

t=5

? You may use a calculator and round your answer to the nearest thousandth.

\newline

Posted 2 months ago

Question

A particle travels along the

x

-axis such that its velocity is given by

v(t)=t^{2.1} \sin (2 t)

. Find all times when the speed of the particle is equal to

2

on the interval

0 \leq t \leq 4

. You may use a calculator and round your answer to the nearest thousandth.

\newline

t=

Posted 2 months ago

Question

A particle travels along the

x

-axis such that its velocity is given by

v(t)=\left(t^{2.3}+5\right) \sin (2 t)

. What is the distance traveled by the particle over the interval

0 \leq t \leq 5

? You may use a calculator and round your answer to the nearest thousandth.

\newline

Posted 2 months ago

Question

A particle travels along the

x

-axis such that its velocity is given by

v(t)=t^{1.9} \cos (3 t-1)

. What is the distance traveled by the particle over the interval

0 \leq t \leq 3

? You may use a calculator and round your answer to the nearest thousandth.

\newline

Posted 2 months ago

Question

A particle travels along the

x

-axis such that its velocity is given by

v(t)=\left(t^{1.6}+3\right) \cos (3 t)

. What is the average velocity of the particle on the interval

0 \leq t \leq 6

? You may use a calculator and round your answer to the nearest thousandth.

\newline

Posted 2 months ago

Question

A particle travels along the

x

-axis such that its acceleration is given by

a(t)=\left(t^{0.5}+3\right) \cos (3 t)

. If the velocity of the particle is

v=-2

t=1

, what is the velocity of the particle when

t=1

? You may use a calculator and round your answer to the nearest thousandth.

\newline

Posted 2 months ago

Question

A particle travels along the

x

-axis such that its velocity is given by

v(t)=t^{2.1} \sin (2 t+5)

. Find all times when the speed of the particle is equal to

1

on the interval

0 \leq t \leq 3

. You may use a calculator and round your answer to the nearest thousandth.

\newline

t=

Posted 2 months ago

Question

A particle travels along the

x

-axis such that its velocity is given by

v(t)=\left(t^{2.1}+5\right) \cos (2 t)

. What is the distance traveled by the particle over the interval

0 \leq t \leq 5

? You may use a calculator and round your answer to the nearest thousandth.

\newline

Posted 2 months ago