Textbook Question
Derivative Calculations
In Exercises 1–12, find the first and second derivatives.
y = x³/3 + x²/2 + x/4
Ch. 3 - Derivatives
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 3, Problem 3.5.35
Tangent Lines
In Exercises 35–38, graph the curves over the given intervals, together with their tangent lines at the given values of x. Label each curve and tangent line with its equation.
y = sin x, −3π/2 ≤ x ≤ 2π
x = −π, 0, 3π/2
Verified step by step guidance1
Identify the function and the points where the tangent lines are to be found. The function is y = sin(x), and the points are x = -π, x = 0, and x = 3π/2.
Find the derivative of the function y = sin(x) to determine the slope of the tangent line at any point x. The derivative is y' = cos(x).
Calculate the slope of the tangent line at each specified point by substituting the x-values into the derivative. For x = -π, x = 0, and x = 3π/2, compute y'(-π), y'(0), and y'(3π/2).
Determine the y-coordinate of the function at each specified x-value by substituting these x-values into the original function y = sin(x). Calculate y(-π), y(0), and y(3π/2).
Use the point-slope form of the equation of a line, y - y₁ = m(x - x₁), where m is the slope and (x₁, y₁) is a point on the line, to write the equation of the tangent line at each specified point. Substitute the slopes and points found in the previous steps to get the equations of the tangent lines.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Tangent Line
A tangent line to a curve at a given point is a straight line that touches the curve at that point without crossing it. The slope of the tangent line is equal to the derivative of the curve's equation at that point. For the function y = sin x, the tangent line at a specific x-value can be found using the derivative, y' = cos x.
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Slopes of Tangent Lines
Derivative of Trigonometric Functions
The derivative of a function provides the rate at which the function's value changes with respect to changes in its input. For trigonometric functions like y = sin x, the derivative is y' = cos x. This derivative is crucial for determining the slope of the tangent line at any point on the curve, which is necessary for graphing the tangent lines.
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Graphing Trigonometric Functions
Graphing trigonometric functions involves plotting their values over a specified interval. For y = sin x, the graph is a wave-like pattern that oscillates between -1 and 1. Understanding the periodic nature and key points, such as x = -π, 0, and 3π/2, helps in accurately plotting the curve and its tangent lines, ensuring each is labeled with its equation.
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Related Practice
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In Exercises 53 and 54, find dr/ds.
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