Textbook Question
{Use of Tech} Midpoint Riemann sums with a calculator Consider the following definite integrals.
(a) Write the midpoint Riemann sum in sigma notation for an arbitrary value of n.
∫₀⁴ (4𝓍― 𝓍²) d𝓍
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Ch. 5 - Integration
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 5, Problem 5.3.111a
Zero net area Consider the function ƒ(𝓍) = 𝓍² ― 4𝓍 .
(a) Graph ƒ on the interval 𝓍 ≥ 0.
Verified step by step guidance1
Step 1: Start by analyzing the given function ƒ(𝓍) = 𝓍² - 4𝓍. Identify its key features, such as the degree of the polynomial (quadratic) and the leading coefficient (positive, indicating the parabola opens upwards).
Step 2: Find the critical points of the function by taking its derivative. Compute ƒ'(𝓍) = d/d𝓍 [𝓍² - 4𝓍] = 2𝓍 - 4. Set ƒ'(𝓍) = 0 to solve for 𝓍, which gives the critical points.
Step 3: Determine the vertex of the parabola. The vertex occurs at 𝓍 = -b/(2a) for a quadratic function in the form ax² + bx + c. Here, a = 1 and b = -4, so the vertex is at 𝓍 = 2. Evaluate ƒ(2) to find the corresponding y-coordinate of the vertex.
Step 4: Identify the x-intercepts by solving ƒ(𝓍) = 0. Factorize the quadratic equation 𝓍² - 4𝓍 = 0 as 𝓍(𝓍 - 4) = 0, which gives the solutions 𝓍 = 0 and 𝓍 = 4. These are the points where the graph crosses the x-axis.
Step 5: Plot the graph of ƒ(𝓍) on the interval 𝓍 ≥ 0. Mark the vertex, x-intercepts, and other key points. Sketch the parabola, ensuring it opens upwards and passes through the identified points.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Graphing Functions
Graphing functions involves plotting points on a coordinate plane to visualize the relationship between the input (x-values) and output (y-values) of a function. For the function ƒ(𝓍) = 𝓍² - 4𝓍, this means calculating y-values for various x-values, particularly within the specified interval x ≥ 0, and connecting these points to form a continuous curve.
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Graph of Sine and Cosine Function
Finding Roots
Finding the roots of a function refers to determining the values of x for which the function equals zero. For ƒ(𝓍) = 𝓍² - 4𝓍, this involves solving the equation 𝓍² - 4𝓍 = 0, which can be factored to find the x-intercepts. These roots are critical for understanding where the graph intersects the x-axis and can indicate changes in the function's behavior.
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Understanding Area Under the Curve
The area under the curve of a function on a given interval can provide insights into the function's behavior, such as net area, which accounts for regions above and below the x-axis. In this case, analyzing the graph of ƒ(𝓍) = 𝓍² - 4𝓍 will help determine if the net area is zero, which occurs when the positive and negative areas cancel each other out within the specified interval.
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05:59Estimating the Area Under a Curve with Right Endpoints & Midpoint
Related Practice
Textbook Question
Suppose ƒ is an odd function, ∫₀⁴ ƒ(𝓍) d𝓍 = 3 , and ∫₀⁸ ƒ(𝓍) d𝓍 = 9 .
(a) Evaluate ∫₋₈⁴ ƒ(𝓍) d𝓍 .
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Textbook Question
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
(a) Consider the linear function ƒ(𝓍) = 2x + 5 and the region bounded by its graph and the x-axis on the interval [3,6]. Suppose the area of this region is approximated using midpoint Riemann sums. Then the approximations give the exact area of the region for any number of subintervals.
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Textbook Question
Approximating displacement The velocity in ft/s of an object moving along a line is given by v = 3t² + 1 on the interval 0 ≤ t ≤ 4, where t is measured in seconds.
(a) Divide the interval [0,4] into n = 4 subintervals, [0,1] , [1.2] , [2,3] , and [3,4]. On each subinterval, assume the object moves at a constant velocity equal to v evaluated at the midpoint of the subinterval, and use these approximations to estimate the displacement of the object on [0, 4] (see part (a) of the figure)
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Textbook Question
Working with area functions Consider the function ƒ and the points a, b, and c.
(a) Find the area function A (𝓍) = ∫ₐˣ ƒ(t) dt using the Fundamental Theorem.
ƒ(𝓍) = sin 𝓍 ; a = 0 , b = π/2 , c = π
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Textbook Question
Symmetry properties Suppose ∫₀⁴ ƒ(𝓍) d𝓍 = 10 and ∫₀⁴ g(𝓍) d𝓍 = 20. Furthermore, suppose ƒ is an even function and g is an odd function. Evaluate the following integrals.
(c) ∫₋₄⁴ (4ƒ(𝓍) ― 3g(𝓍))d𝓍
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