Domanda del libro di testo
21–30. Derivatives
b. Evaluate f'(a) for the given values of a.
f(x) = 4x²+1; a= 2,4
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Ch. 3 - Derivatives
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 3, Problema 3.10.86c
Tangents and inverses Suppose L(x)=ax+b (with a≠0) is the equation of the line tangent to the graph of a one-to-one function f at (x0,y0). Also, suppose M(x)=cx+d is the equation of the line tangent to the graph of f^−1 at (y0,x0).
c. Prove that L^−1(x)=M(x).
Guida verificata passo dopo passo1
Start by understanding the relationship between a function and its inverse. If L(x) is tangent to f at (x0, y0), then the slope of L(x), which is 'a', is equal to the derivative of f at x0, i.e., f'(x0) = a.
For the inverse function f^−1, the derivative at a point (y0, x0) is the reciprocal of the derivative of f at x0. Therefore, the slope of M(x), which is 'c', is equal to 1/a, i.e., c = 1/a.
The line L(x) = ax + b is tangent to f at (x0, y0), so it passes through the point (x0, y0). Therefore, y0 = ax0 + b, which can be rearranged to find b: b = y0 - ax0.
Similarly, the line M(x) = cx + d is tangent to f^−1 at (y0, x0), so it passes through the point (y0, x0). Therefore, x0 = cy0 + d, which can be rearranged to find d: d = x0 - cy0.
To prove L^−1(x) = M(x), find the inverse of L(x). The inverse L^−1(x) is found by swapping x and y in the equation y = ax + b and solving for y. This gives L^−1(x) = (x - b)/a. Substitute b = y0 - ax0 into this equation to show that L^−1(x) = M(x) = (1/a)x + (x0 - (1/a)y0).
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Inverse Functions
An inverse function essentially reverses the effect of the original function. If f(x) takes an input x and produces an output y, then the inverse function f^−1(y) takes y back to x. For a function to have an inverse, it must be one-to-one, meaning each output is produced by exactly one input.
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Inverse Cosine
Tangent Lines
A tangent line to a function at a given point represents the instantaneous rate of change of the function at that point. The slope of the tangent line is given by the derivative of the function at that point. In the context of the problem, the tangent lines L(x) and M(x) represent the slopes of the original function f and its inverse f^−1, respectively.
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05:13Slopes of Tangent Lines
Properties of Derivatives
The derivative of a function at a point provides information about the function's behavior near that point, including its slope. For inverse functions, a key property is that the slopes of the tangent lines at corresponding points are reciprocals of each other. This relationship is crucial for proving that the inverse of the tangent line L(x) equals the tangent line M(x).
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06:21Properties of Functions
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Surface area of a cone The lateral surface area of a cone of radius r and height h (the surface area excluding the base) is A = πr√r²+h².
b. Evaluate this derivative when r=30 and h=40.
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Domanda del libro di testo
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c. Differentiate both sides of the identity sin 2t = 2 sin t cost to prove that cos 2t = cos²t−sin²t.
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Domanda del libro di testo
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b. Evaluate this derivative when a=6 and b=10.
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Domanda del libro di testo
62–65. {Use of Tech} Graphing f and f'
c. Verify that the zeros of f' correspond to points at which f has a horizontal tangent line.
f(x) = (x−1) sin^−1 x on [−1,1]
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Domanda del libro di testo
97–100. Logistic growth Scientists often use the logistic growth function P(t) = P₀K / P₀+(K−P₀)e^−r₀t to model population growth, where P₀ is the initial population at time t=0, K is the carrying capacity, and r₀ is the base growth rate. The carrying capacity is a theoretical upper bound on the total population that the surrounding environment can support. The figure shows the sigmoid (S-shaped) curve associated with a typical logistic model. <IMAGE>
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c. How fast (in fish per year) is the population growing at t=0? At t=5?
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