Ch. 3 - Derivatives

Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook

Hass 15th Edition

Ch. 3 - Derivatives

Problem 3.6.83b

Chapter 3, Problem 3.6.83b

b. Slopes on a tangent curve What is the smallest value the slope of the curve can ever have on the interval −2 < x < 2? Give reasons for your answer.

Verified step by step guidance

First, identify the function for which you need to find the slope of the tangent. The slope of the tangent at any point on a curve is given by the derivative of the function at that point.

Calculate the derivative of the function. This derivative will represent the slope of the tangent line at any point x on the curve.

Determine the critical points of the derivative by setting the derivative equal to zero and solving for x. These points are where the slope could potentially be at a minimum or maximum.

Evaluate the derivative at the critical points and also at the endpoints of the interval (x = -2 and x = 2) to find the slope values. Since the interval is open, consider the behavior of the derivative as x approaches these endpoints.

Compare the slope values obtained from the critical points and the endpoints to determine the smallest slope value on the interval -2 < x < 2. This will be the minimum slope of the tangent on the given interval.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Derivative

The derivative of a function at a point provides the slope of the tangent line to the curve at that point. It is a fundamental tool in calculus for understanding how a function changes. In this context, finding the smallest slope involves determining the minimum value of the derivative over the given interval.

Critical Points

Critical points occur where the derivative of a function is zero or undefined. These points are potential candidates for local minima or maxima. To find the smallest slope on the interval, one must evaluate the derivative at critical points and endpoints to identify the minimum value.

Interval Analysis

Interval analysis involves examining the behavior of a function within a specific range of x-values. For the interval −2 < x < 2, it is crucial to analyze the derivative's behavior to determine where it reaches its minimum value, considering both critical points and the interval's boundaries.

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