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36-second reel · English

Six verbs, one textbook

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To answer, we shrink the interval. As h tends to zero, the secant line turns into the tangent line, and its slope becomes the instantaneous rate of change.

Calculus Volume 1.pdf✓ Read · 769 pages

What are we studying?

The whole source
Chapter 1 · Functions and Graphspp. 15–112
Chapter 2 · Limitspp. 113–194
Chapter 3 · Derivativespp. 195–302
Chapter 4 · Applications of Derivativespp. 303–444

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Study planCalculus: Derivatives (Chapter 3)10 phases

01 Foundations of Derivatives02 The Derivative as a Function03 Differentiation Rules04 Derivatives as Rates of Change

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f′(a) = limh→0 f(a + h) − f(a)h

h = 0.00slope ≈ 0.64

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Question 11 of 11Score 10 / 11

What does the slope of the tangent line to a function f(x) at a specific point (a, f(a)) represent?

The average rate of change over a large interval
The instantaneous rate of change of the function at that pointCorrect!
The total area accumulated under the curve up to that point
The difference between the maximum and minimum values of the function

As stated on p. 204, the slope of a tangent line and instantaneous velocity are related concepts that both measure the instantaneous rate of change of a function at a specific point.

Remember.

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Question

What is the relationship between a function being differentiable at a point and being continuous at that point?

Show answer

Answer

If a function is differentiable at a point, it is continuous there. The converse is not always true.

↺ Again✓ Got it

Deck complete!

10Known on the first try2Needed repeats

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Answering from: Calculus Volume 1

Explain implicit differentiation with an example from the book

…involves differentiating both sides of the equation with respect to x, while treating y as a function of x (applying the chain rule), and then solving the resulting equation for dy/dx (p. 296).

Example (p. 278):

dydx = 6y − 3x²3y² − 6x = 2y − x²y² − 2x

Chapter 3 · Derivatives296

3.8 Implicit Differentiation

Implicit differentiation takes three steps. First, differentiate both sides of the equation with respect to x, treating y as a function of x and applying the chain rule. Then solve the result for dy/dx.

dydx = 2y − x²y² − 2x

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