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

3. Techniques of Differentiation

Derivatives of Trig Functions

Calculus

3. Techniques of Differentiation

Derivatives of Trig Functions

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2:48 minutes

Problem 3.5.55b

Textbook Question

An object oscillates along a vertical line, and its position in centimeters is given by y(t) = 30(sint - 1), where t ≥ 0 is measured in seconds and y is positive in the upward direction.
Find the velocity of the oscillator, v(t) =y′(t).

Verified step by step guidance

To find the velocity of the oscillator, we need to differentiate the position function y(t) = 30(sin(t) - 1) with respect to time t.

Recall that the derivative of sin(t) with respect to t is cos(t). Therefore, apply the derivative to each term in the function y(t).

The derivative of the constant term -30 is 0, since constants do not change with respect to t.

Differentiate the term 30sin(t) using the chain rule. The derivative of 30sin(t) is 30cos(t).

Combine the derivatives to get the velocity function v(t) = y'(t) = 30cos(t).

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Key Concepts

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

Differentiation

Differentiation is a fundamental concept in calculus that involves finding the derivative of a function. The derivative represents the rate of change of the function with respect to its variable. In this context, differentiating the position function y(t) = 30(sin(t) - 1) will yield the velocity function v(t), which describes how the position of the object changes over time.

Trigonometric Functions

Trigonometric functions, such as sine and cosine, are essential in modeling periodic phenomena, including oscillations. The sine function, in particular, describes the oscillatory behavior of the object in this problem. Understanding the properties of these functions, including their ranges and periodicity, is crucial for analyzing the motion of the oscillator.

Velocity

Velocity is a vector quantity that indicates the rate of change of an object's position with respect to time. In this scenario, the velocity function v(t) is derived from the position function y(t) by differentiation. It provides insight into how fast and in which direction the object is moving along the vertical line, which is vital for understanding its oscillatory motion.

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23–51. Calculating derivatives Find the derivative of the following functions.

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23–51. Calculating derivatives Find the derivative of the following functions.

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Find the derivative of the following functions.

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Textbook Question

An object oscillates along a vertical line, and its position in centimeters is given by y(t)=30(sint - 1), where t ≥ 0 is measured in seconds and y is positive in the upward direction.

At what times and positions is the velocity zero?

Textbook Question

An object oscillates along a vertical line, and its position in centimeters is given by y(t) = 30(sin t - 1), where t ≥ 0 is measured in seconds and y is positive in the upward direction.

The acceleration of the oscillator is a(t) = v′(t). Find and graph the acceleration function.

Textbook Question

Verifying derivative formulas Verify the following derivative formulas using the Quotient Rule.

d/dx (sec x) = sec x tan x

Textbook Question

Verifying derivative formulas Verify the following derivative formulas using the Quotient Rule.

d/dx (csc x) = -csc x cot x

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An object oscillates along a vertical line, and its position in centimeters is given by y(t) = 30(sint - 1), where t ≥ 0 is measured in seconds and y is positive in the upward direction.Find the velocity of the oscillator, v(t) =y′(t).

Key Concepts

Differentiation

Trigonometric Functions

Velocity

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An object oscillates along a vertical line, and its position in centimeters is given by y(t) = 30(sint - 1), where t ≥ 0 is measured in seconds and y is positive in the upward direction.
Find the velocity of the oscillator, v(t) =y′(t).