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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2:21 minutes

Problem 3.5.66

Textbook Question

Evaluate the following limits or state that they do not exist. (Hint: Identify each limit as the derivative of a function at a point.)
lim x→π/2 cos x/x−π/2

Verified step by step guidance

Recognize that the given limit can be interpreted as the derivative of a function at a specific point. The expression lim x→π/2 (cos x)/(x−π/2) resembles the definition of a derivative.

Recall the definition of the derivative of a function f at a point a: f'(a) = lim x→a (f(x) - f(a))/(x - a).

Identify the function f(x) = cos x and the point a = π/2. Notice that f(a) = cos(π/2) = 0.

Rewrite the limit in the form of the derivative definition: lim x→π/2 (cos x - cos(π/2))/(x - π/2). This matches the derivative definition for f(x) = cos x at x = π/2.

Conclude that the limit represents the derivative of f(x) = cos x at x = π/2, which is f'(π/2). To find this derivative, compute f'(x) = -sin x and evaluate it at x = π/2.

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Key Concepts

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

Limits

A limit is a fundamental concept in calculus that describes the behavior of a function as its input approaches a certain value. It helps in understanding how functions behave near points of interest, including points where they may not be explicitly defined. Evaluating limits is crucial for determining continuity, derivatives, and integrals.

Derivatives

The derivative of a function at a point measures the rate at which the function's value changes as its input changes. It is defined as the limit of the average rate of change of the function over an interval as the interval approaches zero. In this context, recognizing the limit as a derivative allows for the application of derivative rules to evaluate the limit.

L'Hôpital's Rule

L'Hôpital's Rule is a method for evaluating limits that result in indeterminate forms, such as 0/0 or ∞/∞. It states that if the limit of f(x)/g(x) leads to an indeterminate form, the limit can be found by taking the derivative of the numerator and the derivative of the denominator separately. This rule is particularly useful in simplifying complex limit problems.

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