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

Derivatives as Functions

Calculus

2. Intro to Derivatives

Derivatives as Functions

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Multiple Choice

Given the graph of a function $f (x)$ , which of the following statements best describes the graph of its derivative $f^{'} (x)$ ?

The graph of

f^{'} (x)

is positive where

f (x)

is increasing and negative where

f (x)

is decreasing.

The graph of

f^{'} (x)

is identical to the graph of

f (x)

but shifted up by one unit.

The graph of

f^{'} (x)

is always above the

x

-axis if

f (x)

is concave up.

The graph of

f^{'} (x)

is negative where

f (x)

is increasing and positive where

f (x)

is decreasing.

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Verified step by step guidance

Step 1: Understand the relationship between a function f(x) and its derivative f'(x). The derivative f'(x) represents the rate of change or slope of the function f(x) at any given point.

Step 2: Recall that when f(x) is increasing (its slope is positive), the derivative f'(x) will be positive. Conversely, when f(x) is decreasing (its slope is negative), the derivative f'(x) will be negative.

Step 3: Analyze the incorrect options: The graph of f'(x) is not identical to f(x) shifted up by one unit, as the derivative represents the slope, not a vertical shift. Similarly, f'(x) is not always above the x-axis when f(x) is concave up; concavity relates to the second derivative, not the first.

Step 4: Evaluate the correct statement: The graph of f'(x) is positive where f(x) is increasing and negative where f(x) is decreasing. This aligns with the definition of the derivative as the slope of the function.

Step 5: Conclude that understanding the behavior of f'(x) in relation to f(x) is crucial for interpreting the graph of the derivative. The correct statement is based on the fundamental relationship between a function and its derivative.