Ch. 3 - Derivatives

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 3 - Derivatives

Problem 3.2.73c

Chapter 3, Problem 3.2.73c

Vertical tangent lines If a function f is continuous at a and lim x→a| f′(x)|=∞, then the curve y=f(x) has a vertical tangent line at a, and the equation of the tangent line is x=a. If a is an endpoint of a domain, then the appropriate one-sided derivative (Exercises 71–72) is used. Use this information to answer the following questions.
73. {Use of Tech} Graph the following functions and determine the location of the vertical tangent lines.
c. f(x) = √|x-4|

Verified step by step guidance

Step 1: Understand the problem. We need to find the location of vertical tangent lines for the function f(x) = \(\sqrt{|x-4|}\). A vertical tangent line occurs where the derivative of the function approaches infinity.

Step 2: Analyze the function. The function f(x) = \(\sqrt{|x-4|}\) is continuous everywhere in its domain, which is all real numbers. However, the expression inside the square root, |x-4|, changes behavior at x = 4.

Step 3: Find the derivative of the function. To find where the vertical tangent line occurs, we need to compute the derivative f'(x). Start by rewriting the function as f(x) = (|x-4|)^{1/2}. Use the chain rule and the derivative of the absolute value function to find f'(x).

Step 4: Evaluate the behavior of the derivative at x = 4. Since the absolute value function has a corner at x = 4, check the limit of the derivative as x approaches 4 from both sides. Specifically, calculate \(\lim\)_{x \(\to\) 4^-} f'(x) and \(\lim\)_{x \(\to\) 4^+} f'(x).

Step 5: Determine the location of the vertical tangent line. If either of the one-sided limits from Step 4 approaches infinity, then there is a vertical tangent line at x = 4. The equation of this vertical tangent line is x = 4.

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Key Concepts

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

Vertical Tangent Lines

A vertical tangent line occurs at a point on a curve where the slope of the tangent approaches infinity. This typically indicates that the derivative of the function is undefined or infinite at that point. In the context of calculus, if the limit of the absolute value of the derivative approaches infinity as x approaches a certain value, the function has a vertical tangent line at that point.

Continuity of Functions

A function is continuous at a point if there are no breaks, jumps, or holes in the graph at that point. For a function to have a vertical tangent line, it must first be continuous at the point of interest. This means that the function's value at that point is well-defined, and the behavior of the function around that point can be analyzed using limits.

One-Sided Derivatives

One-sided derivatives are used to analyze the behavior of a function at endpoints or points where the function may not be differentiable in the traditional sense. The left-hand derivative considers the slope of the tangent as you approach the point from the left, while the right-hand derivative does so from the right. In cases where a function is defined only on one side of a point, one-sided derivatives provide crucial information about the function's behavior at that point.

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One-Sided Limits

Related Practice

Textbook Question

{Use of Tech} Power and energy Power and energy are often used interchangeably, but they are quite different. Energy is what makes matter move or heat up. It is measured in units of joules or Calories, where 1 Cal=4184 J. One hour of walking consumes roughly 10⁶J, or 240 Cal. On the other hand, power is the rate at which energy is used, which is measured in watts, where 1 W=1 J/s. Other useful units of power are kilowatts (1 kW=10³ W) and megawatts (1 MW=10⁶ W). If energy is used at a rate of 1 kW for one hour, the total amount of energy used is 1 kilowatt-hour (1 kWh=3.6×10⁶ J) Suppose the cumulative energy used in a large building over a 24-hr period is given by E(t)=100t+4t²−t³ / 9kWh where t=0 corresponds to midnight.

c. Graph the power function and interpret the graph. What are the units of power in this case?

Textbook Question

Derivatives of inverse functions from a table Use the following tables to determine the indicated derivatives or state that the derivative cannot be determined. <IMAGE>

c. (f^-1)'(1)

Textbook Question

62–65. {Use of Tech} Graphing f and f'

c. Verify that the zeros of f' correspond to points at which f has a horizontal tangent line.

f(x)=(sec^−1 x)/x on [1,∞)

Textbook Question

{Use of Tech} Spring oscillations 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) = 10sin t−10cos t, where x is positive when the mass is above the equilibrium position. <IMAGE>

c. At what times is the velocity of the mass zero?

Textbook Question

Derivatives of sin^n x Calculate the following derivatives using the Product Rule.

c. d/dx (sin⁴ x)

Textbook Question

Computing the derivative of f(x) = e^-x

c. Use parts (a) and (b) to find the derivative of f(x) = e^-x.