Textbook Question
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. Assume ƒ and ƒ' are continuous functions for all real numbers.
(c) ∫ₐᵇ ƒ'(𝓍) d𝓍 = ƒ(b) ―ƒ(a) .
4
views
8. Definite Integrals
Fundamental Theorem of Calculus
2:11 minutesProblem 5.3.87b
Textbook Question
Matching functions with area functions Match the functions ƒ, whose graphs are given in a― d, with the area functions A (𝓍) = ∫₀ˣ ƒ(t) dt, whose graphs are given in A–D.
Verified step by step guidance1
Step 1: Understand the relationship between the function ƒ(t) and the area function A(x). The area function A(x) = ∫₀ˣ ƒ(t) dt represents the accumulated area under the curve of ƒ(t) from t = 0 to t = x. The shape of A(x) depends on the behavior of ƒ(t).
Step 2: Analyze the graph of ƒ(t) in image (b). The function ƒ(t) is a linear function that starts positive at t = 0, decreases linearly, crosses the t-axis, and becomes negative. This indicates that the area accumulated initially increases, then decreases as the negative contributions dominate.
Step 3: Match the behavior of ƒ(t) to the corresponding area function A(x). Since ƒ(t) starts positive, A(x) initially increases. As ƒ(t) crosses the t-axis and becomes negative, A(x) starts decreasing. This matches the graph labeled (A), which shows an initial increase followed by a decrease.
Step 4: Verify the match by considering the slope of A(x). The slope of A(x) is given by ƒ(t). In graph (A), the slope of A(x) is positive initially (matching the positive ƒ(t)), becomes zero at the peak (where ƒ(t) = 0), and then becomes negative (matching the negative ƒ(t)).
Step 5: Conclude that the graph of ƒ(t) in image (b) corresponds to the area function A(x) in graph (A). This is based on the consistent relationship between the function and its integral.
Verified video answer for a similar problem:This video solution was recommended by our tutors as helpful for the problem above
Video duration:
2mWas this helpful?
Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Definite Integral
A definite integral represents the signed area under the curve of a function f(t) from a specified lower limit to an upper limit. It is denoted as A(x) = ∫₀ˣ f(t) dt, where A(x) gives the accumulated area from 0 to x. Understanding this concept is crucial for relating the function f(t) to its area function A(x).
Recommended video:
05:43
Definition of the Definite Integral
Area Function
An area function A(x) is defined as the integral of a function f(t) from a lower limit to x. It provides a way to visualize how the area under the curve changes as x varies. The shape and behavior of the area function graph can reveal important characteristics about the original function f(t), such as its increasing or decreasing nature.
Recommended video:
05:06Finding Area When Bounds Are Not Given
Graphical Interpretation
Graphical interpretation involves analyzing the visual representation of functions and their integrals. For the functions f(t) and their corresponding area functions A(x), understanding how the graph of f(t) influences the graph of A(x) is essential. For instance, if f(t) is positive, A(x) will increase, while if f(t) is negative, A(x) will decrease, illustrating the relationship between the two.
Recommended video:
05:02Determining Differentiability Graphically
6:11mWatch next
Master Fundamental Theorem of Calculus Part 1 with a bite sized video explanation from Patrick
Start learningRelated Videos
Related Practice