Ch. 3 - Derivatives

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 3 - Derivatives

Problem 3.8.64b

Chapter 3, Problem 3.8.64b

Vertical tangent lines
b. Does the curve have any horizontal tangent lines? Explain.

Verified step by step guidance

To determine if the curve has any horizontal tangent lines, we need to find where the derivative of the function is equal to zero. Horizontal tangent lines occur at points where the slope of the curve is zero.

First, identify the function that describes the curve. Let's denote this function as f(x).

Next, compute the derivative of the function, f'(x), using differentiation rules. This derivative represents the slope of the tangent line at any point x on the curve.

Set the derivative f'(x) equal to zero and solve for x. This will give you the x-values where the slope of the tangent line is zero, indicating potential horizontal tangent lines.

Finally, verify these x-values by substituting them back into the original function f(x) to find the corresponding y-values. This will confirm the points on the curve where horizontal tangent lines occur.

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Key Concepts

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

Tangent Lines

Tangent lines are straight lines that touch a curve at a single point without crossing it. The slope of the tangent line at a point on the curve represents the instantaneous rate of change of the function at that point. Understanding tangent lines is crucial for analyzing the behavior of curves, particularly in determining where they are increasing or decreasing.

Vertical Tangent Lines

A vertical tangent line occurs when the slope of the tangent approaches infinity, which typically happens when the derivative of the function is undefined at that point. This can indicate a cusp or a vertical asymptote in the curve. Identifying vertical tangents is important for understanding the limits and behavior of the function near those points.

Horizontal Tangent Lines

A horizontal tangent line occurs when the slope of the tangent line is zero, indicating that the function has a local maximum or minimum at that point. To find horizontal tangents, one must set the derivative of the function equal to zero and solve for the corresponding x-values. This concept is essential for analyzing critical points and the overall shape of the curve.

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Related Practice

Textbook Question

21–30. Derivatives

b. Evaluate f'(a) for the given values of a.

f(x) = 4x²+1; a= 2,4

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Textbook Question

{Use of Tech} A mixing tank A 500-liter (L) tank is filled with pure water. At time t=0, a salt solution begins flowing into the tank at a rate of 5 L/min. At the same time, the (fully mixed) solution flows out of the tank at a rate of 5.5 L/min. The mass of salt in grams in the tank at any time t≥0 is given by M(t) = 250(1000−t)(1−10−³⁰(1000−t)¹⁰) and the volume of solution in the tank is given by V(t) = 500-0.5t.

b. Graph the volume function and verify that the tank is empty when t=1000 min.

Textbook Question

Deriving trigonometric identities

b. Verify that you obtain the same identity for sin2t as in part (a) if you differentiate the identity cos 2t = 2 cos² t−1.

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Textbook Question

58–59. Carry out the following steps.

b. Find the slope of the curve at the given point.

xy^5/2+x^3/2y=12; (4, 1)

Textbook Question

13-26 Implicit differentiation Carry out the following steps.

b. Find the slope of the curve at the given point.