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

For the function $f (x)$ , in which direction(s) does the derivative $f^{'} (x)$ provide information about the behavior of $f (x)$ ?

The derivative describes the instantaneous rate of change of

f (x)

with respect to

y

, which is the vertical direction.

The derivative does not provide directional information about

f (x)

The derivative describes the instantaneous rate of change of

f (x)

with respect to

x

, which is the horizontal direction.

The derivative provides information about both the horizontal and vertical directions simultaneously.

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Verified step by step guidance

Step 1: Understand the concept of a derivative. The derivative of a function, f'(x), represents the instantaneous rate of change of the function f(x) with respect to its input variable x. This means it tells us how f(x) changes as x changes.

Step 2: Clarify the relationship between the derivative and the horizontal direction. Since the derivative f'(x) is calculated with respect to x, it provides information about the behavior of f(x) along the horizontal axis (the x-axis).

Step 3: Clarify the relationship between the derivative and the vertical direction. The value of f'(x) itself does not directly describe the vertical direction (the y-axis). Instead, it indicates how steeply f(x) is increasing or decreasing, which indirectly affects the vertical change in f(x).

Step 4: Address the misconception about simultaneous directional information. The derivative does not provide information about both horizontal and vertical directions simultaneously. It specifically describes the rate of change of f(x) with respect to x, which is tied to the horizontal direction.

Step 5: Conclude that the correct interpretation is: The derivative describes the instantaneous rate of change of f(x) with respect to x, which is the horizontal direction.