Rates of Change and Monotonic Functions

Name: Rates of Change and Monotonic Functions
Author: StemPy Lab

The derivative tells how fast one quantity changes when another changes. Use it to follow a moving particle and a changing cost. Solve related rates with a ladder, a balloon, a filling cone and a shadow. Then find where a function increases or decreases from the sign of its derivative. It covers the first part of the Indian syllabus unit (NCERT, JEE Main) and the matching topics of A-Level, IB and AP.

3 lessons, about 1 h 15 min3 simulations10 problems3 practice questionsPython in every lesson

Created by StemPy Lab

Course outline (3 lessons)

Lesson 1about 25 min
The derivative as a rate of change
The derivative $\dfrac{dy}{dx}$ is the rate at which $y$ changes when $x$ changes, and it carries the units of $y$ per unit of $x$ . An average rate uses two points. The derivative is its limit as the two points meet. It gives the speed of a moving object, the rate at which an area grows with its radius, and a marginal cost.
1 simulation1 practice question1 Python program2 problems1 Python lab
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Lesson 2about 25 min
Related rates
In a related rates problem, two or more quantities change with time and are tied together by an equation. Differentiate that equation with respect to $t$ using the chain rule, and then put in the numbers for the moment you need. The steps are the same for a sliding ladder, an inflating balloon, a filling cone and a moving shadow.
1 simulation1 practice question1 Python program5 problems
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Lesson 3about 25 min
Increasing and decreasing functions
A function is increasing on an interval if its graph rises as you move to the right, and decreasing if it falls. The sign of the derivative tells which. If $f'(x) > 0$ on an interval, $f$ is strictly increasing there. If $f'(x) < 0$ , it is strictly decreasing. To find the intervals, solve $f'(x) = 0$ and test the sign of $f'$ in each gap. A function that only increases or only decreases is called monotonic.
1 simulation1 practice question1 Python program3 problems1 Python lab1 question set1 note
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Rates of Change and Monotonic Functions: Practice Paper

15 questions · 20 min

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Rates of Change and Monotonic Functions

Course outline (3 lessons)

The derivative as a rate of change

Related rates

Increasing and decreasing functions

Test Your Learning

Your teacher

Discussion (0)