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Chapter 3: Problem 5

Given a function \(f\) and a point \(a\) in its domain, what does \(f^{\prime}(a)\) represent?

Short Answer

Expert verified
Answer: \(f^{\prime}(a)\) represents the rate of change of the function \(f\) at the point \(a\), which is the slope of the tangent line to the graph of the function at the point (\(a\), \(f(a)\)). It shows how quickly the function is increasing or decreasing at that particular point.

Step by step solution

01

Recall the concept of derivatives

A derivative is a measure of how a function changes with respect to its input variable, often denoted by \(f^{\prime}(x)\). The derivative of a function at a specific point \(a\) in the domain indicates the rate of change or slope of the function at that precise point.
02

Geometric interpretation of the derivative

Geometrically, the derivative of a function \(f\) at a point \(a\) is the slope of the tangent line to the graph of the function at the point (\(a\), \(f(a)\)). This means that when we graph the function, the value of \(f^{\prime}(a)\) gives us the steepness of the tangent line at that point.
03

Understand the meaning of \(f^{\prime}(a)\)

The value of \(f^{\prime}(a)\) represents the rate of change of the function \(f\) at the point \(a\). In other words, it shows how quickly the function is increasing or decreasing at that particular point. When the derivative is positive, it means the function is increasing, and when it is negative, it means the function is decreasing. A zero derivative at a point indicates that the function has a local maximum, minimum, or a horizontal tangent at that point.

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Most popular questions from this chapter

General logarithmic and exponential derivatives Compute the following derivatives. Use logarithmic differentiation where appropriate. $$\frac{d}{d x}\left(1+\frac{1}{x}\right)^{x}$$

A woman attached to a bungee cord jumps from a bridge that is \(30 \mathrm{m}\) above a river. Her height in meters above the river \(t\) seconds after the jump is \(y(t)=15\left(1+e^{-t} \cos t\right),\) for \(t \geq 0\). a. Determine her velocity at \(t=1\) and \(t=3\). b. Use a graphing utility to determine when she is moving downward and when she is moving upward during the first 10 s. c. Use a graphing utility to estimate the maximum upward velocity.

A spring hangs from the ceiling at equilibrium with a mass attached to its end. Suppose you pull downward on the mass and release it 10 inches below its equilibrium position with an upward push. The distance \(x\) (in inches) of the mass from its equilibrium position after \(t\) seconds is given by the function \(x(t)=10 \sin t-10 \cos t,\) where \(x\) is positive when the mass is above the equilibrium position. a. Graph and interpret this function. b. Find \(\frac{d x}{d t}\) and interpret the meaning of this derivative. c. At what times is the velocity of the mass zero? d. The function given here is a model for the motion of an object on a spring. In what ways is this model unrealistic?

Use the properties of logarithms to simplify the following functions before computing \(f^{\prime}(x)\). $$f(x)=\ln \frac{(2 x-1)(x+2)^{3}}{(1-4 x)^{2}}$$

General logarithmic and exponential derivatives Compute the following derivatives. Use logarithmic differentiation where appropriate. $$\frac{d}{d x}(2 x)^{2 x}$$.
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