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Chapter 3: Problem 6

Explain the relationships among the slope of a tangent line, the instantaneous rate of change, and the value of the derivative at a point.

Short Answer

Expert verified
Question: Explain the relationships among the slope of a tangent line, the instantaneous rate of change, and the value of the derivative at a point. Answer: The slope of a tangent line at a given point on a function is equal to the instantaneous rate of change of the function at that point, which represents how steeply the function's value is changing at that point. The instantaneous rate of change is also equal to the value of the derivative at that point, which is the result of differentiating the function with respect to its input variable and evaluating the resulting expression for the desired input value. Therefore, at any given point, the slope of the tangent line, the instantaneous rate of change, and the value of the derivative are all interconnected and equal to each other.

Step by step solution

01

Defining Slope of a Tangent Line

The slope of a tangent line to a curve at a specific point is the rate at which the curve is changing at that point. It can be thought of as the steepness of a straight line that "touches" the curve at that point without crossing over it. Mathematically, the slope is represented by the rise (change in y) divided by the run (change in x) between any two points on the line.
02

Defining Instantaneous Rate of Change

The instantaneous rate of change is the rate at which a quantity is changing at a specific moment in time. In the context of a function, it represents the speed at which the value of the function is changing at a specific point and can be found by taking the derivative of the function with respect to its input variable.
03

Defining Value of the Derivative at a Point

The value of the derivative at a point is the result of differentiating a function with respect to its input variable and then evaluating the resulting expression for the input value. It is essentially the instantaneous rate of change at the specified point on the function.
04

Connection between Slope of a Tangent Line and Instantaneous Rate of Change

The slope of a tangent line at a specific point on a function represents how steeply the function's value is changing at that point, which is the instantaneous rate of change. Thus, we can say that the slope of a tangent line to a curve at a given point is equal to the instantaneous rate of change of the curve at that point.
05

Connection between Instantaneous Rate of Change and Value of the Derivative

The instantaneous rate of change is the result of differentiating a function with respect to its input variable and then evaluating the resulting expression for the desired input value. As the value of the derivative represents the rate at which the function's value is changing at a given point, the instantaneous rate of change of a function at a specific point is equal to the value of its derivative at that point.
06

Conclusion: Relationships between the Three Concepts

The slope of a tangent line, the instantaneous rate of change, and the value of the derivative at a point are all interconnected ideas. The slope of a tangent line at a given point on a function is equal to the instantaneous rate of change, which in turn is equal to the value of the derivative at that point. Therefore, at any given point we have: Slope of Tangent Line = Instantaneous Rate of Change of the function = Value of the Derivative at the point

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Most popular questions from this chapter

Prove the following identities and give the values of \(x\) for which they are true. $$\cos \left(\sin ^{-1} x\right)=\sqrt{1-x^{2}}$$

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