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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Multiple Choice

Which of the following is the period of the function $z = \sin (x - y)$ with respect to $x$ ?

π

There is no period

2 π

1

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Verified step by step guidance

Step 1: Recall the definition of the period of a function. The period of a function is the smallest positive value T such that the function repeats itself, i.e., f(x + T) = f(x).

Step 2: Consider the given function z = sin(x - y). The sine function has a fundamental period of 2π, meaning it repeats every 2π units.

Step 3: Observe that the variable y does not affect the periodicity with respect to x because it is treated as a constant when analyzing the function's behavior in terms of x.

Step 4: Substitute x with (x + 2π) in the function: z = sin((x + 2π) - y). Simplify the argument of the sine function: (x + 2π - y) = (x - y) + 2π.

Step 5: Use the property of the sine function, sin(a + 2π) = sin(a), to confirm that z = sin(x - y) repeats itself every 2π with respect to x. Therefore, the period of the function with respect to x is 2π.