/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 43 Given the function \(f,\) find t... [FREE SOLUTION] | 91影视

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Chapter 3: Problem 43

Given the function \(f,\) find the slope of the line tangent to the graph of \(f^{-1}\) at the specified point on the graph of \(f^{-1}\) $$f(x)=\sqrt{x}$$

Short Answer

Expert verified
Solution: The slope of the tangent line at the point (2, 4) is 4.

Step by step solution

01

Find the derivative of \(f(x)\)

The given function is \(f(x)=\sqrt{x}\), so to find its derivative, we can use the power rule: $$f'(x) = \frac{d}{dx} x^{\frac{1}{2}} = \frac{1}{2}x^{-\frac{1}{2}}$$
02

Identify the inverse function derivative formula

Given an invertible function \(f(x)\), the derivative of its inverse function, \(f^{-1}(x)\), is found using the formula: $$(f^{-1})'(x) = \frac{1}{f'(f^{-1}(x))}$$
03

Evaluate the derivative of the inverse function at the specified point

The problem asks us to find the slope of the tangent line at the point (2,4) on the graph of \(f^{-1}(x)\). From step 1, we have: $$f'(x) = \frac{1}{2}x^{-\frac{1}{2}}$$ From Step 2, we can write the derivative of the inverse function as: $$(f^{-1})'(x) = \frac{1}{f'(f^{-1}(x))}$$ Now, we need to evaluate this expression at x = 2. However, we have the specified point in terms of the graph of \(f^{-1}(x)\). Since \((2, 4)\) is on the graph of \(f^{-1}(x)\), then \((4, 2)\) is on the graph of \(f(x)\): $$f(4) = 2$$ Now we can compute the derivative of \(f^{-1}(x)\) at \(x=2\): $$(f^{-1})'(2) = \frac{1}{f'(f^{-1}(2))} = \frac{1}{f'(4)}$$ Using the expression for \(f'(x)\), compute \(f'(4)\): $$f'(4) = \frac{1}{2}(4)^{-\frac{1}{2}} = \frac{1}{4}$$ Finally, find the slope of the tangent line at the specified point on the graph of the inverse function: $$(f^{-1})'(2) = \frac{1}{f'(4)} = \frac{1}{\frac{1}{4}} = 4$$ The slope of the tangent line to the graph of \(f^{-1}(x)\) at point (2,4) is 4.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Tangent Line
A tangent line is a straight line that just "touches" the curve at a given point, without crossing over it. You can think of it as just grazing the surface. In calculus, finding the slope of this line is crucial because it helps us understand the behavior of the function at that specific point. Imagine you have a smooth hill and you place a plank of wood touching it at one point. The plank represents the tangent line. The slope of this tangent line tells you how steep the hill is at that exact spot. To get the slope of the tangent line to a function at a point, you need to calculate the derivative of the function at that point. For the inverse function, this involves using the inverse derivative formula. Key points:
  • The tangent line is a straight line touching the curve at one point only.
  • The slope of a tangent line indicates how the function is behaving at that precise location.
  • To find the slope of the tangent to an inverse function, apply the inverse derivative formula.
Derivative
Derivatives are a cornerstone of calculus, and they tell us how a function behaves as its inputs change. Think of a derivative as a tool that reveals the "rate of change" of a function at any point.To understand derivatives, imagine driving a car and checking your speedometer. Even as your speed varies, the speedometer gives you an immediate rate of change of your position over time. This rate of change at any given instant is what the derivative calculates for functions.The derivative of a function is denoted by the symbol \(f'(x)\), and it's obtained mathematically through various rules including the power rule, which simplifies the process of finding derivatives of polynomial expressions.
  • Derivatives signify the rate of change of a function.
  • Calculated using symbols like \(f'(x)\).
  • Essential for understanding the behavior of curves and calculating tangents.
Power Rule
The power rule is a simple yet powerful tool for finding the derivative of functions that are polynomials. It's expressed as follows: If you have a function \(f(x) = x^n\), then its derivative \(f'(x) = nx^{n-1}\).This rule covers any function where \(x\) is raised to a constant power. For example, if \(f(x) = x^3\), the derivative is \(f'(x) = 3x^2\). By multiplying the power by the coefficient of \(x\) and reducing the power by one, you find the derivative easily.In our exercise, we used it to differentiate \(f(x)=\sqrt{x}\), which is \(x^{1/2}\). Applying the power rule yields \(f'(x) = \frac{1}{2}x^{-1/2}\).
  • Applies to functions where \(x\) is raised to a power.
  • Simplifies finding derivatives for polynomial functions.
  • Essential in solving calculus problems efficiently.

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Most popular questions from this chapter

Proof of \(\frac{d}{d x}(\cos x)=-\sin x\) Use the definition of the derivative and the trigonometric identity $$ \cos (x+h)=\cos x \cos h-\sin x \sin h $$ to prove that \(\frac{d}{d x}(\cos x)=-\sin x\)

Electrostatic force The magnitude of the electrostatic force between two point charges \(Q\) and \(q\) of the same sign is given by \(F(x)=\frac{k Q q}{x^{2}},\) where \(x\) is the distance (measured in meters) between the charges and \(k=9 \times 10^{9} \mathrm{N} \cdot \mathrm{m}^{2} / \mathrm{C}^{2}\) is a physical constant (C stands for coulomb, the unit of charge; N stands for newton, the unit of force). a. Find the instantaneous rate of change of the force with respect to the distance between the charges. b. For two identical charges with \(Q=q=1 \mathrm{C},\) what is the instantaneous rate of change of the force at a separation of \(x=0.001 \mathrm{m} ?\) c. Does the magnitude of the instantaneous rate of change of the force increase or decrease with the separation? Explain.

Proof by induction: derivative of \(e^{k x}\) for positive integers \(k\) Proof by induction is a method in which one begins by showing that a statement, which involves positive integers, is true for a particular value (usually \(k=1\) ). In the second step, the statement is assumed to be true for \(k=n\), and the statement is proved for \(k=n+1,\) which concludes the proof. a. Show that \(\frac{d}{d x}\left(e^{k x}\right)=k e^{k x}\) for \(k=1\) b. Assume the rule is true for \(k=n\) (that is, assume \(\left.\frac{d}{d x}\left(e^{n x}\right)=n e^{n x}\right),\) and show this implies that the rule is true for \(k=n+1 .\) (Hint: Write \(e^{(n+1) x}\) as the product of two functions, and use the Product Rule.)

Witch of Agnesi Let \(y\left(x^{2}+4\right)=8\) (see figure). a. Use implicit differentiation to find \(\frac{d y}{d x}\) b. Find equations of all lines tangent to the curve \(y\left(x^{2}+4\right)=8\) when \(y=1\) c. Solve the equation \(y\left(x^{2}+4\right)=8\) for \(y\) to find an explicit expression for \(y\) and then calculate \(\frac{d y}{d x}\) d. Verify that the results of parts (a) and (c) are consistent.

Vertical tangent lines a. Determine the points where the curve \(x+y^{2}-y=1\) has a vertical tangent line (see Exercise 53 ). b. Does the curve have any horizontal tangent lines? Explain.
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