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

2. Intro to Derivatives

Basic Graphing of the Derivative

Calculus

2. Intro to Derivatives

Basic Graphing of the Derivative

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Multiple Choice

Given the graph of the function $f (x)$ below, which of the following graphs best represents its derivative $f^{'} (x)$ ?

A graph that is negative where

f (x)

is increasing and positive where

f (x)

is decreasing.

A graph that is positive where

f (x)

is increasing, negative where

f (x)

is decreasing, and zero at the local maxima and minima of

f (x)

A graph that is identical to

f (x)

A graph that is always positive regardless of the behavior of

f (x)

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Verified step by step guidance

Step 1: Recall the relationship between a function f(x) and its derivative f'(x). The derivative f'(x) represents the slope of the tangent line to the graph of f(x) at any given point. This means f'(x) tells us how f(x) is changing at each point.

Step 2: Analyze the behavior of f(x) on the graph. Identify intervals where f(x) is increasing (positive slope), decreasing (negative slope), and where it has local maxima or minima (slope is zero).

Step 3: Understand that when f(x) is increasing, the slope of the tangent line is positive, so f'(x) will be positive in these intervals. Similarly, when f(x) is decreasing, the slope of the tangent line is negative, so f'(x) will be negative in these intervals.

Step 4: At the local maxima and minima of f(x), the tangent line is horizontal, meaning the slope is zero. Therefore, f'(x) will be zero at these points.

Step 5: Based on this analysis, the graph of f'(x) will be positive where f(x) is increasing, negative where f(x) is decreasing, and zero at the local maxima and minima of f(x). This matches the description of the correct answer provided in the problem.