Ch. 5 - Integration

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch. 5 - Integration

Problem 5.3.111a

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 guidance

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.

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.

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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Related Practice

Textbook Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. Assume ƒ, ƒ', and ƒ'' are continuous functions for all real numbers.

(a) ∫ ƒ(𝓍) ƒ'(𝓍) d𝓍 = ½ (ƒ(𝓍))² + C.

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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) = cos 𝓍 ; a = 0 , b = π/2 , c = π

Textbook Question

Mass from density A thin 10-cm rod is made of an alloy whose density varies along its length according to the function shown in the figure. Assume density is measured in units of g/cm. In Chapter 6, we show that the mass of the rod is the area under the density curve.

(a) Find the mass of the left half of the rod (0 ≤ x ≤ 5) .

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Textbook Question

Approximating areas Estimate the area of the region bounded by the graph of ƒ(𝓍) = x² + 2 and the x-axis on [0, 2] in the following ways.

(a) Divide [0, 2] into n = 4 subintervals and approximate the area of the region using a left Riemann sum. Illustrate the solution geometrically.

Textbook Question

Sigma notation Evaluate the following expressions.

(a) 10

∑ κ

κ=1

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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Zero net area Consider the function ƒ(𝓍) = 𝓍² ― 4𝓍 .(a) Graph ƒ on the interval 𝓍 ≥ 0.

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Key Concepts

Graphing Functions

Finding Roots

Understanding Area Under the Curve

Zero net area Consider the function ƒ(𝓍) = 𝓍² ― 4𝓍 .
(a) Graph ƒ on the interval 𝓍 ≥ 0.