Ch. 3 - Derivatives

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

Hass 15th Edition

Ch. 3 - Derivatives

Problem 3.88

Chapter 3, Problem 3.88

In Exercises 83–88, find equations for the lines that are tangent, and the lines that are normal, to the curve at the given point.

x³/² + 2y³/² = 17, (1, 4)

Verified step by step guidance

First, understand that to find the tangent line to the curve at a given point, we need to determine the derivative of the curve with respect to x, which will give us the slope of the tangent line at that point.

Given the equation of the curve:

x^{\frac{3}{2}} + 2 y^{\frac{3}{2}} = 17

, we will use implicit differentiation to find

\frac{d}{y} \frac{d}{x}

Differentiate both sides of the equation with respect to x. Remember to apply the chain rule when differentiating terms involving y, as y is a function of x. This will yield:

\frac{3}{2} x^{\frac{1}{2}} + 2 y^{\frac{1}{2}} \frac{3}{2} \frac{d}{y} \frac{d}{x} = 0

Solve the resulting equation for

\frac{d}{y} \frac{d}{x}

to find the slope of the tangent line at the point (1, 4). Substitute x = 1 and y = 4 into the differentiated equation to find the specific slope at that point.

Once the slope of the tangent line is found, use the point-slope form of a line,

y - y_1 = m(x - x_1)

, where m is the slope and (x_1, y_1) is the point (1, 4), to write the equation of the tangent line. For the normal line, use the negative reciprocal of the tangent slope and apply the same point-slope formula.

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Key Concepts

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

Implicit Differentiation

Implicit differentiation is a technique used to find the derivative of a function when it is not explicitly solved for one variable in terms of another. In this problem, the equation x^(3/2) + 2y^(3/2) = 17 involves both x and y, requiring implicit differentiation to find dy/dx, which is essential for determining the slope of the tangent line at the given point.

Tangent Line

The tangent line to a curve at a given point is a straight line that touches the curve at that point without crossing it. Its slope is equal to the derivative of the curve at that point. For the equation x^(3/2) + 2y^(3/2) = 17 at (1, 4), the tangent line's equation can be found using the slope obtained from implicit differentiation and the point-slope form of a line.

Normal Line

The normal line to a curve at a given point is perpendicular to the tangent line at that point. Its slope is the negative reciprocal of the tangent line's slope. Once the slope of the tangent line is determined, the normal line's equation can be derived using the point-slope form, ensuring it passes through the same point on the curve.

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Slopes of Tangent Lines

Related Practice

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If r = sin(f(t)), f(0) = π/3, and f'(0) = 4, then what is dr/dt at t = 0?

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Textbook Question

Approximation Error

In Exercises 29–34, each function f(x) changes value when x changes from x₀ to x₀ + dx. Find

a. the change Δf = f(x₀ + dx) − f(x₀);

b. the value of the estimate df = fʹ(x₀) dx; and

c. the approximation error |Δf − df|.

f(x) = x⁻¹, x₀ = 0.5, dx = 0.1

Textbook Question

Slopes on the graph of the tangent function Graph y = tan x and its derivative together on (−π/2, π/2). Does the graph of the tangent function appear to have a smallest slope? A largest slope? Is the slope ever negative? Give reasons for your answers.

Textbook Question

Find the derivatives of the functions in Exercises 1–42.

𝔂 = ( √ x )²

( 1 + x )

Textbook Question

Normal lines to a parabola Show that if it is possible to draw three normal lines from the point (a, 0) to the parabola x = y² shown in the accompanying diagram, then a must be greater than 1/2. One of the normal lines is the x-axis. For what value of a are the other two normal lines perpendicular?

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In Exercises 83–88, find equations for the lines that are tangent, and the lines that are normal, to the curve at the given point.x³/² + 2y³/² = 17, (1, 4)

Verified video answer for a similar problem:

Key Concepts

Implicit Differentiation

Tangent Line

Normal Line

In Exercises 83–88, find equations for the lines that are tangent, and the lines that are normal, to the curve at the given point.

x³/² + 2y³/² = 17, (1, 4)