5. Let f (t) = 2t – 4. %3D А. Sketch or provide the graph of the function f on the interval [0,5]. Given A(x) = S ƒ(t)dt, evaluate the following. Explain: i. A(4) ii. А(3) iii. A' (4) B.
5. Let f (t) = 2t – 4. %3D А. Sketch or provide the graph of the function f on the interval [0,5]. Given A(x) = S ƒ(t)dt, evaluate the following. Explain: i. A(4) ii. А(3) iii. A' (4) B.
Glencoe Algebra 1, Student Edition, 9780079039897, 0079039898, 2018
18th Edition
ISBN:9780079039897
Author:Carter
Publisher:Carter
Chapter1: Expressions And Functions
Section: Chapter Questions
Problem 74SGR
Related questions
Question
![### Problem 5
**Let \( f(t) = 2t - 4 \).**
**A.** Sketch or provide the graph of the function \( f \) on the interval \([0, 5]\).
*Explanation of Graph*: The function \( f(t) = 2t - 4 \) is a linear function with a slope of 2 and a y-intercept of -4. On the interval \([0, 5]\), at \( t = 0 \), \( f(t) = -4 \). At \( t = 5 \), \( f(t) = 2(5) - 4 = 6 \). So, the graph is a straight line starting from the point (0, -4) to (5, 6).
**B.** Given \( A(x) = \int_{3}^{x} f(t) \, dt \), evaluate the following. Explain:
**i.** \( A(4) \)
**Explanation**: Calculate the definite integral of \( f(t) = 2t - 4 \) from 3 to 4:
\[
A(4) = \int_{3}^{4} (2t - 4) \, dt.
\]
Solve the integral to find the area under the curve from \( t = 3 \) to \( t = 4 \).
**ii.** \( A(3) \)
**Explanation**: Since the integral is evaluated from 3 to 3:
\[
A(3) = \int_{3}^{3} (2t - 4) \, dt = 0.
\]
The area under the curve from a point to itself is zero.
**iii.** \( A'(4) \)
**Explanation**: According to the Fundamental Theorem of Calculus, the derivative of the integral function with respect to x, \( A'(x) \), at a point is equal to the value of the original function \( f(t) \) at that point.
\[
A'(4) = f(4) = 2(4) - 4 = 8 - 4 = 4.
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd1b542d2-f99f-41f7-9f8d-fbf0665f8261%2Fa79f1811-5b3a-41f7-b396-ca69d0cac846%2Fnpuvt5_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem 5
**Let \( f(t) = 2t - 4 \).**
**A.** Sketch or provide the graph of the function \( f \) on the interval \([0, 5]\).
*Explanation of Graph*: The function \( f(t) = 2t - 4 \) is a linear function with a slope of 2 and a y-intercept of -4. On the interval \([0, 5]\), at \( t = 0 \), \( f(t) = -4 \). At \( t = 5 \), \( f(t) = 2(5) - 4 = 6 \). So, the graph is a straight line starting from the point (0, -4) to (5, 6).
**B.** Given \( A(x) = \int_{3}^{x} f(t) \, dt \), evaluate the following. Explain:
**i.** \( A(4) \)
**Explanation**: Calculate the definite integral of \( f(t) = 2t - 4 \) from 3 to 4:
\[
A(4) = \int_{3}^{4} (2t - 4) \, dt.
\]
Solve the integral to find the area under the curve from \( t = 3 \) to \( t = 4 \).
**ii.** \( A(3) \)
**Explanation**: Since the integral is evaluated from 3 to 3:
\[
A(3) = \int_{3}^{3} (2t - 4) \, dt = 0.
\]
The area under the curve from a point to itself is zero.
**iii.** \( A'(4) \)
**Explanation**: According to the Fundamental Theorem of Calculus, the derivative of the integral function with respect to x, \( A'(x) \), at a point is equal to the value of the original function \( f(t) \) at that point.
\[
A'(4) = f(4) = 2(4) - 4 = 8 - 4 = 4.
\]
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 3 steps with 1 images
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.Recommended textbooks for you
Glencoe Algebra 1, Student Edition, 9780079039897…
Algebra
ISBN:
9780079039897
Author:
Carter
Publisher:
McGraw Hill
Trigonometry (MindTap Course List)
Trigonometry
ISBN:
9781337278461
Author:
Ron Larson
Publisher:
Cengage Learning
Glencoe Algebra 1, Student Edition, 9780079039897…
Algebra
ISBN:
9780079039897
Author:
Carter
Publisher:
McGraw Hill
Trigonometry (MindTap Course List)
Trigonometry
ISBN:
9781337278461
Author:
Ron Larson
Publisher:
Cengage Learning
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
Intermediate Algebra
Algebra
ISBN:
9781285195728
Author:
Jerome E. Kaufmann, Karen L. Schwitters
Publisher:
Cengage Learning
College Algebra (MindTap Course List)
Algebra
ISBN:
9781305652231
Author:
R. David Gustafson, Jeff Hughes
Publisher:
Cengage Learning