Skip to main content
Ch. 3 - Derivatives
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 3 - DerivativesProblem 3.7.39
Chapter 3, Problem 3.7.39

Slopes, Tangent Lines, and Normal Lines


In Exercises 31–40, verify that the given point is on the curve and find the lines that are (a) tangent and (b) normal to the curve at the given point.


y = 2 sin(πx – y), (1,0)

Verified step by step guidance
1
First, verify that the point (1, 0) is on the curve by substituting x = 1 and y = 0 into the equation y = 2 sin(πx - y). Check if the left-hand side equals the right-hand side.
To find the tangent line, we need the derivative of the function. Differentiate the equation y = 2 sin(πx - y) implicitly with respect to x. Remember to apply the chain rule and implicit differentiation.
Solve the resulting equation for dy/dx, which represents the slope of the tangent line at any point (x, y) on the curve.
Substitute the point (1, 0) into the expression for dy/dx to find the slope of the tangent line at this specific point.
The normal line is perpendicular to the tangent line. Use the negative reciprocal of the tangent slope to find the slope of the normal line. Then, use the point-slope form of a line equation to write the equations of both the tangent and normal lines at the point (1, 0).

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
6m
Was this helpful?

Key Concepts

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

Point Verification on a Curve

To verify that a given point is on a curve, substitute the point's coordinates into the curve's equation. If the equation holds true, the point lies on the curve. For the equation y = 2 sin(πx - y), substitute x = 1 and y = 0 to check if the left side equals the right side.
Recommended video:
11:41
Summary of Curve Sketching

Tangent Line to a Curve

The tangent line to a curve at a given point is a straight line that touches the curve at that point without crossing it. To find the tangent line, calculate the derivative of the curve's equation to determine the slope at the given point, then use the point-slope form of a line to write the equation of the tangent line.
Recommended video:
05:13
Slopes of Tangent Lines

Normal Line to a Curve

The normal line to a curve at a given point is perpendicular to the tangent line at that point. To find the normal line, first determine the slope of the tangent line, then use the negative reciprocal of this slope for the normal line. Apply the point-slope form using the given point to write the equation of the normal line.
Recommended video:
05:13
Slopes of Tangent Lines
Related Practice
Textbook Question

Derivative Calculations


In Exercises 1–12, find the first and second derivatives.


y = x² + x + 8

Textbook Question

Slopes and Tangent Lines


In Exercises 1–4, use the grid and a straight edge to make a rough estimate of the slope of the curve (in y-units per x-unit) at the points P₁ and P₂.

Textbook Question

If L = √(x² + y²), dx/dt = –1, and dy/dt = 3, find dL/dt when x = 5 and y = 12.

Textbook Question

Cylinder pressure If gas in a cylinder is maintained at a constant temperature T, the pressure P is related to the volume V by a formula of the form

P = (nRT / (V − nb)) − (an² / V²),

in which a, b, n, and R are constants. Find dP/dV. (See accompanying figure.)


<IMAGE>

Textbook Question

Find the derivatives of the functions in Exercises 19–40.


g(t) = (1 + sin(3t) / (3 − 2t))⁻¹

Textbook Question

In Exercises 29 and 30, find the slope of the curve at the given points.


(x² + y²)² = (x – y)² at (1,0) and (1,–1)