Ch. 4 - Applications of the Derivative

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 4 - Applications of the Derivative

Problem 4.5.25

Chapter 4, Problem 4.5.25

Minimum distance Find the point P on the line y = 3x that is closest to the point (50, 0). What is the least distance between P and (50, 0)?

Verified step by step guidance

Identify the line equation y = 3x and the point (50, 0) from which we need to find the minimum distance to a point P on the line.

Express the coordinates of point P on the line as (x, 3x) since any point on the line y = 3x can be represented in this form.

Use the distance formula to express the distance D between point P(x, 3x) and the point (50, 0): D = sqrt((x - 50)^2 + (3x - 0)^2).

Simplify the distance formula: D = sqrt((x - 50)^2 + 9x^2). This simplifies to D = sqrt(10x^2 - 100x + 2500).

To find the minimum distance, minimize the expression under the square root, 10x^2 - 100x + 2500, by finding its derivative, setting it to zero, and solving for x. This will give the x-coordinate of point P that minimizes the distance.

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Key Concepts

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

Distance Formula

The distance formula calculates the distance between two points in a Cartesian plane. For points (x1, y1) and (x2, y2), the distance d is given by d = √((x2 - x1)² + (y2 - y1)²). This formula is essential for determining how far the point P on the line is from the point (50, 0).

Line Equation

The equation of a line describes the relationship between x and y coordinates of points on that line. In this case, the line is given by y = 3x, which indicates that for every unit increase in x, y increases by three units. Understanding this equation helps in identifying the coordinates of point P that lies on the line.

Optimization

Optimization in calculus involves finding the maximum or minimum values of a function. In this problem, we need to minimize the distance from point P on the line to the point (50, 0). This typically involves using techniques such as taking derivatives and setting them to zero to find critical points.

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Related Practice

Textbook Question

Explain the Mean Value Theorem with a sketch.

Textbook Question

{Use of Tech} Tumor size In a study conducted at Dartmouth College, mice with a particular type of cancerous tumor were treated with the chemotherapy drug Cisplatin. If the volume of one of these tumors at the time of treatment is V₀, then the volume of the tumor t days after treatment is modeled by the function V(t) = V₀ (0.99e⁻⁰·¹²¹⁶ᵗ + 0.01e⁰·²³⁹ᵗ). (Source: Undergraduate Mathematics for the Life Sciences, MAA Notes No. 81, 2013)

Plot a graph of y = 0.99e⁻⁰·¹²¹⁶ᵗ + 0.01e⁰·²³⁹ᵗ, for 0 ≤ t ≤ 16, and describe the tumor size over time. Use Newton’s method to determine when the tumor decreases to half of its original size.

Textbook Question

Differentials Consider the following functions and express the relationship between a small change in x and the corresponding change in y in the form dy = f'(x)dx.

f(x) = ln (1 - x)

Textbook Question

Particular antiderivatives For the following functions f, find the antiderivative F that satisfies the given condition.

f(v) = sec v tan v; F(0) = 2, -π/2 < v < π/2

Textbook Question

Give the antiderivatives of xᵖ . For what values of p does your answer apply?

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Textbook Question

Locating critical points Find the critical points of the following functions. Assume a is a nonzero constant.

ƒ(x) = x² √(x + 5)

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