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Ch. 3 - Derivatives
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 3 - DerivativesProblem 56a
Chapter 3, Problem 56a

Assume that a particle’s position on the x-axis is given by


x = 3 cos t + 4 sin t,


where x is measured in feet and t is measured in seconds.


a. Find the particle’s position when t = 0, t = π/2, and t = π.

Verified step by step guidance
1
To find the particle's position at a specific time, substitute the given values of t into the position function x = 3 cos t + 4 sin t.
For t = 0, substitute t = 0 into the equation: x = 3 cos(0) + 4 sin(0). Recall that cos(0) = 1 and sin(0) = 0.
For t = π/2, substitute t = π/2 into the equation: x = 3 cos(π/2) + 4 sin(π/2). Recall that cos(π/2) = 0 and sin(π/2) = 1.
For t = π, substitute t = π into the equation: x = 3 cos(π) + 4 sin(π). Recall that cos(π) = -1 and sin(π) = 0.
Evaluate each expression to find the particle's position at t = 0, t = π/2, and t = π using the trigonometric values provided.

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Key Concepts

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

Trigonometric Functions

Trigonometric functions, such as sine and cosine, are fundamental in describing periodic phenomena. In this context, they represent the particle's oscillatory motion along the x-axis. Understanding how to evaluate these functions at specific angles, like 0, π/2, and π, is crucial for determining the particle's position at given times.
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Introduction to Trigonometric Functions

Evaluating Trigonometric Expressions

To find the particle's position at specific times, we need to evaluate the trigonometric expression x = 3 cos t + 4 sin t. This involves substituting the given values of t into the expression and calculating the result. Familiarity with the unit circle and the values of sine and cosine at key angles is essential for this process.
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Introduction to Trigonometric Functions

Position Function

The position function x = 3 cos t + 4 sin t describes the particle's location on the x-axis over time. It combines the effects of two harmonic motions, each with its amplitude and phase. Understanding how to interpret and manipulate this function allows us to predict the particle's position at any given time t.
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Relations and Functions
Related Practice
Textbook Question

Quadratic approximations


[Technology Exercise] e. Find the quadratic approximation to h(x) = √(1 + x) at x = 0. Graph h and its quadratic approximation together. Comment on what you see.

Textbook Question

Quadratic approximations


[Technology Exercise] c. Graph f(x) = 1/(1 − x) and its quadratic approximation at x = 0. Then zoom in on the two graphs at the point (0,1). Comment on what you see.

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Textbook Question

[Technology Exercise]


Graph y = 1/(2√x) in a window that has 0 ≤ x ≤ 2. Then, on the same screen, graph

y = (√(x + h) − √x)/h

for h = 1, 0.5, 0.1. Then try h = −1, −0.5, −0.1. Explain what is going on.

Textbook Question

Find the value of dy/dt at t = 0 if y = 3 sin 2x and x = t² + π.

Textbook Question

Assume that a particle’s position on the x-axis is given by


x = 3 cos t + 4 sin t,


where x is measured in feet and t is measured in seconds.


b. Find the particle’s velocity when t = 0, t = π/2, and t = π.

Textbook Question

Quadratic approximations


d. Find the quadratic approximation to g(x) = 1/x at x = 1. Graph g and its quadratic approximation together. Comment on what you see