Domanda del libro di testo
Use linear approximations to estimate the following quantities. Choose a value of a to produce a small error.
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Ch. 4 - Applications of the Derivative
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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TOC.tableOfContents
- 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: 9780136847243Non è quello che usi tu?Cambia libro di testo
Capitolo 4, Problema 4.1.73
Trajectory high point A stone is launched vertically upward from a cliff 192 ft above the ground at a speed of 64 ft/s. Its height above the ground t seconds after the launch is given by s = -16t² + 64t + 192, for 0 ≤ t ≤ 6. When does the stone reach its maximum height?
Guida verificata passo dopo passo1
Identify the function that represents the height of the stone above the ground: \( s(t) = -16t^2 + 64t + 192 \). This is a quadratic function, which represents a parabola.
Recognize that the maximum height of the stone corresponds to the vertex of the parabola. For a quadratic function \( ax^2 + bx + c \), the vertex occurs at \( t = -\frac{b}{2a} \).
Substitute the values of \( a \) and \( b \) from the function \( s(t) = -16t^2 + 64t + 192 \) into the vertex formula: \( a = -16 \) and \( b = 64 \).
Calculate the time \( t \) at which the stone reaches its maximum height using the formula: \( t = -\frac{64}{2(-16)} \).
Verify that the calculated time \( t \) is within the given interval \( 0 \leq t \leq 6 \) to ensure it is a valid solution for the problem.
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Quadratic Functions
The height of the stone is modeled by a quadratic function, which is a polynomial of degree two. Quadratic functions have a parabolic shape and can be expressed in the form s(t) = at² + bt + c. The coefficients a, b, and c determine the direction and position of the parabola, with 'a' indicating whether it opens upwards or downwards.
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6:04
Introduction to Polynomial Functions
Vertex of a Parabola
The maximum height of the stone corresponds to the vertex of the parabola represented by the quadratic function. For a downward-opening parabola (where a < 0), the vertex can be found using the formula t = -b/(2a). This point gives the time at which the maximum height occurs, which is crucial for solving the problem.
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7:42Properties of Parabolas
Calculus and Optimization
Calculus provides tools for finding maximum and minimum values of functions, which is essential in this context. By taking the derivative of the height function and setting it to zero, we can find critical points that indicate where the maximum height occurs. This process is part of optimization, a key application of calculus in real-world scenarios.
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10:13Intro to Applied Optimization: Maximizing Area
Pratica correlata
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Functions from derivatives Use the derivative f' to determine the x-coordinates of the local maxima and minima of f, and the intervals on which f is increasing or decreasing. Sketch a possible graph of f (f is not unique).
f'(x) = 10 sin 2x on [-2π, 2π]
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Domanda del libro di testo
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Domanda del libro di testo
23–68. Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation.
∫ (3x + 1) (4 - x) dx
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Domanda del libro di testo
Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_x→ 0 (eˣ - 1) / (x² + 3x)
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Domanda del libro di testo
{Use of Tech} Finding roots with Newton’s method For the given function f and initial approximation x₀, use Newton’s method to approximate a root of f. Stop calculating approximations when two successive approximations agree to five digits to the right of the decimal point after rounding. Show your work by making a table similar to that in Example 1.
f(x) = tan x - 2x; x₀ = 1.2
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