Table of contents

0. Functions7h 52m
1. Limits and Continuity2h 2m
2. Intro to Derivatives1h 33m
3. Techniques of Differentiation3h 18m
4. Applications of Derivatives2h 38m
5. Graphical Applications of Derivatives6h 2m
6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m
7. Antiderivatives & Indefinite Integrals1h 26m
8. Definite Integrals4h 44m
9. Graphical Applications of Integrals2h 27m
- Area Between Curves
  1h 18m
- Introduction to Volume & Disk Method
  1h 8m
10. Physics Applications of Integrals 3h 16m
- Kinematics
  1h 13m
- Work
  2h 3m
11. Integrals of Inverse, Exponential, & Logarithmic Functions2h 34m
12. Techniques of Integration7h 39m
13. Intro to Differential Equations2h 55m
14. Sequences & Series5h 36m
15. Power Series2h 19m
- Introduction to Power Series
  1h 9m
- Taylor Series & Taylor Polynomials
  1h 10m
16. Parametric Equations & Polar Coordinates7h 58m

0. Functions

Introduction to Trigonometric Functions

Calculus

0. Functions

Introduction to Trigonometric Functions

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1:24 minutes

Problem 55a

Textbook Question

A weight is attached to a spring and reaches its equilibrium position (x = 0). It is then set in motion resulting in a displacement of x = 10 cos t, where x is measured in centimeters and t is measured in seconds. See the accompanying figure.

Find the spring’s displacement when t = 0, t = π/3, and t = 3π/4.

Verified step by step guidance

Identify the displacement function given in the problem: x(t) = 10 cos(t). This function describes the displacement of the spring from its equilibrium position over time.

To find the displacement at a specific time, substitute the given time value into the displacement function. Start with t = 0: x(0) = 10 cos(0).

Evaluate the cosine function at t = 0. Recall that cos(0) = 1, so x(0) = 10 * 1.

Next, substitute t = π/3 into the displacement function: x(π/3) = 10 cos(π/3).

Evaluate the cosine function at t = π/3. Recall that cos(π/3) = 1/2, so x(π/3) = 10 * 1/2.

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Key Concepts

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

Equilibrium Position

The equilibrium position of a spring is the point where the net force acting on the spring is zero. In this context, it is represented by x = 0, where the spring is neither compressed nor stretched. When a weight is attached and set in motion, it oscillates around this equilibrium point, demonstrating simple harmonic motion.

Simple Harmonic Motion (SHM)

Simple Harmonic Motion is a type of periodic motion where an object moves back and forth around an equilibrium position. The displacement of the object can be described by a cosine or sine function, such as x = 10 cos(t), indicating that the motion is sinusoidal and the object oscillates with a specific amplitude and period.

Trigonometric Functions

Trigonometric functions, such as cosine and sine, are fundamental in describing oscillatory motion. In this case, the displacement of the spring is given by x = 10 cos(t), where the cosine function determines the position of the spring at any time t. Understanding these functions is crucial for calculating the displacement at specific time intervals.

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