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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2:19 minutes

Problem 56b

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.

Verified step by step guidance

To find the particle's velocity, we need to determine the derivative of the position function x(t) with respect to time t. The position function is given as x(t) = 3 cos(t) + 4 sin(t).

Differentiate x(t) with respect to t. The derivative of cos(t) is -sin(t), and the derivative of sin(t) is cos(t). Therefore, the derivative of x(t) is v(t) = -3 sin(t) + 4 cos(t).

Now, substitute t = 0 into the velocity function v(t) = -3 sin(t) + 4 cos(t) to find the velocity at t = 0.

Next, substitute t = π/2 into the velocity function v(t) = -3 sin(t) + 4 cos(t) to find the velocity at t = π/2.

Finally, substitute t = π into the velocity function v(t) = -3 sin(t) + 4 cos(t) to find the velocity at t = π.

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Key Concepts

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

Derivative

The derivative of a function represents the rate of change of the function with respect to a variable. In this context, the derivative of the position function x(t) with respect to time t gives the velocity of the particle. Calculating the derivative allows us to determine how the position changes over time, which is essential for finding the velocity at specific time points.

Trigonometric Derivatives

Understanding the derivatives of trigonometric functions is crucial here, as the position function involves sine and cosine. The derivative of cos(t) is -sin(t), and the derivative of sin(t) is cos(t). Applying these rules to the position function x = 3 cos t + 4 sin t helps us find the velocity function, which is necessary to evaluate the velocity at given times.

Evaluating Functions at Specific Points

Once the velocity function is derived, it must be evaluated at specific time points: t = 0, t = π/2, and t = π. This involves substituting these values into the velocity function to find the particle's velocity at these moments. This step is crucial for understanding the particle's motion at different times and requires careful substitution and simplification.

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In Exercises 27–32, find dp/dq.

p = (sin q + cos q) / cos q

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In Exercises 27–32, find dp/dq.

p = (q sin q) / (q² − 1)

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Consider the function

f(x) = { x² cos(2/x), x ≠ 0

0, x = 0

b. Determine f' for x ≠ 0.

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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.

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

Textbook Question

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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

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Find all points on the curve y = tan x, −π/2 < x < π/2, where the tangent line is parallel to the line y = 2x. Sketch the curve and tangent lines together, labeling each with its equation.

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In Exercises 47 and 48, find an equation for

(a) the tangent line to the curve at P

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Textbook Question

In Exercises 47 and 48, find an equation for

(b) the horizontal tangent line to the curve at Q.

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