Midwest Temperature The figure shows the temperature values (in ^∘ F ) on a typical May day in

Name: Problem 29, Chapter 3: Calculus Concepts
Uploaded: 2021-11-30T05:53:43-08:00
Duration: 1 min 7 s
Channel: Carson Merrill

step 1 For these problems, we're going to be looking a little bit closer on how to calculate average ra

Question

Midwest Temperature The figure shows the temperature values (in ${ }^{\circ} \mathrm{F}$ ) on a typical May day in a certain Midwestern city.The equation of the graph is $$ t(x)=-0.8 x^{2}+11.6 x+38.2^{\circ} \mathrm{F} $$ where $x$ is the number of hours since 6 A.M. a. Write the formula for $t^{\prime}$. b. How quickly is the temperature changing at $10 \mathrm{~A} . \mathrm{M} . ?$ c. What is the instantaneous rate of change of the temperature at $4 \mathrm{PM} . ?$ d. Draw and label tangent lines depicting the results from parts $b$ and $c$

   Midwest Temperature The figure shows the temperature values (in ${ }^{\circ} \mathrm{F}$ ) on a typical May day in a certain Midwestern city.The equation of the graph is $$ t(x)=-0.8 x^{2}+11.6 x+38.2^{\circ} \mathrm{F} $$ where $x$ is the number of hours since 6 A.M.
a. Write the formula for $t^{\prime}$.
b. How quickly is the temperature changing at $10 \mathrm{~A} . \mathrm{M} . ?$
c. What is the instantaneous rate of change of the temperature at $4 \mathrm{PM} . ?$
d. Draw and label tangent lines depicting the results from parts $b$ and $c$

Calculus Concepts

Donald R. LaTorre,… 5th Edition

Chapter 3, Problem 29

Instant Answer

Solved by Expert Carson Merrill

11/30/2021

Step 1

The derivative of a function gives us the rate of change of the function at any given point. The derivative of $t(x)$, denoted as $t'(x)$, can be found using the power rule for differentiation, which states that the derivative of $x^n$ is $n*x^{n-1}$. Show more…

Show all steps

Thanks for your feedback!

Midwest Temperature The figure shows the temperature values (in ${ }^{\circ} \mathrm{F}$ ) on a typical May day in a certain Midwestern city.The equation of the graph is $$ t(x)=-0.8 x^{2}+11.6 x+38.2^{\circ} \mathrm{F} $$ where $x$ is the number of hours since 6 A.M. a. Write the formula for $t^{\prime}$. b. How quickly is the temperature changing at $10 \mathrm{~A} . \mathrm{M} . ?$ c. What is the instantaneous rate of change of the temperature at $4 \mathrm{PM} . ?$ d. Draw and label tangent lines depicting the results from parts $b$ and $c$

Play audio

Feedback

Carson Merrill and 87 other subject Calculus 1 / AB educators are ready to help you.

Ask a new question

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Differentiation of Polynomial Functions

This concept involves applying the rules of differentiation to functions that are polynomials. In the context of the question, the temperature is modeled by a quadratic function, and finding its derivative requires using the power rule to compute the rate at which temperature changes over time.

Instantaneous Rate of Change

The instantaneous rate of change is the value of the derivative at a specific point. It represents how fast the dependent variable is changing at a particular moment, analogous to finding the speed at a specific time in real-world applications such as the changing temperature in the given scenario.

Tangent Lines

A tangent line to a curve at a given point is a straight line that just touches the curve at that point, having the same slope as the curve. Drawing and labeling tangent lines helps visualize the instantaneous rate of change and understand the behavior of the function near that specific point.

Please give Ace some feedback