Textbook Question
Displacement from a velocity graph Consider the velocity function for an object moving along a line (see figure).
(a) Describe the motion of the object over the interval [0,6].
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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 Ζ(π) = πΒ² β 4π .
(a) Graph Ζ on the interval π β₯ 0.
Verified step by step guidance1
Step 1: Start by analyzing the given function Ζ(π) = πΒ² - 4π. 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 Ζ'(π) = d/dπ [πΒ² - 4π] = 2π - 4. Set Ζ'(π) = 0 to solve for π, which gives the critical points.
Step 3: Determine the vertex of the parabola. The vertex occurs at π = -b/(2a) for a quadratic function in the form axΒ² + bx + c. Here, a = 1 and b = -4, so the vertex is at π = 2. Evaluate Ζ(2) to find the corresponding y-coordinate of the vertex.
Step 4: Identify the x-intercepts by solving Ζ(π) = 0. Factorize the quadratic equation πΒ² - 4π = 0 as π(π - 4) = 0, which gives the solutions π = 0 and π = 4. These are the points where the graph crosses the x-axis.
Step 5: Plot the graph of Ζ(π) on the interval π β₯ 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 Ζ(π) = πΒ² - 4π, this means calculating y-values for various x-values, particularly within the specified interval x β₯ 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 Ζ(π) = πΒ² - 4π, this involves solving the equation πΒ² - 4π = 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 Ζ(π) = πΒ² - 4π 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
Suppose Ζ is an odd function, β«ββ΄ Ζ(π) dπ = 3 , and β«ββΈ Ζ(π) dπ = 9 .
(a) Evaluate β«βββ΄ Ζ(π) dπ .
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Textbook Question
Free fall On October 14, 2012, Felix Baumgartner stepped off a balloon capsule at an altitude of almost 39 km above Earthβs surface and began his free fall. His velocity in m/s during the fall is given in the figure. It is claimed that Felix reached the speed of sound 34 seconds into his fall and that he continued to fall at supersonic speed for 30 seconds. (Source: http://www.redbullstratos.com)
(a) Divide the interval [34, 64] into n = 5 subintervals with the gridpoints xβ = 34 , xβ = 40 , xβ = 46 , xβ = 52 , xβ = 58 , and xβ = 64. Use left and right Riemann sums to estimate how far Felix fell while traveling at supersonic speed.
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Textbook Question
The velocity in ft/s of an object moving along a line is given by v = Ζ(t) on the interval 0 β€ t β€ 6 (see figure), where t is measured in seconds.
(a) Divide the interval [0,6] into n = 3 subintervals, [0,2] , [2,4] and [4,6]. On each subinterval, assume the object moves at a constant velocity equal to the value of v evaluated at the right endpoint of the subinterval, and use these approximations to estimate the displacement of the object on [0,6] (see part (a) of the figure)
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Textbook Question
Sigma notation Evaluate the following expressions.
(a) 10
β ΞΊ
ΞΊ=1
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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) If Ζ is a constant function on the interval [a,b], then the right and left Riemann sums give the exact value of β«βα΅ Ζ(π) dπ, for any positive integer n.
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