Higher Derivatives and the Mean Value Theorems

Name: Higher Derivatives and the Mean Value Theorems
Author: StemPy Lab

The first derivative is the gradient of a graph, and the second derivative tells how that gradient changes. Find second and higher derivatives, including those of implicit and parametric functions. Then meet Rolle's theorem and the Mean Value Theorem, and see what they say about speed, roots and inequalities. It covers the Indian syllabus (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
Second and higher derivatives
The second derivative is the derivative of the derivative. It tells how fast the slope of a graph is changing. Where $f''$ is positive the graph bends up, and where it is negative the graph bends down. For a parametric curve, $\dfrac{d^2y}{dx^2}$ is the derivative of $\dfrac{dy}{dx}$ with respect to $t$ , divided by $\dfrac{dx}{dt}$ .
1 simulation1 practice question1 Python program4 problems1 Python lab
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Lesson 2about 25 min
Rolle's theorem
Rolle's theorem says that if $f$ is continuous on $[a, b]$ , differentiable on $(a, b)$ and $f(a) = f(b)$ , then $f'(c) = 0$ for some $c$ in $(a, b)$ . The graph has a horizontal tangent somewhere between two points of equal height. The three conditions are all needed. The theorem shows that between two roots of a function lies a root of its derivative.
1 simulation1 practice question1 Python program3 problems
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Lesson 3about 25 min
The Mean Value Theorem and its uses
The Mean Value Theorem says that if $f$ is continuous on $[a, b]$ and differentiable on $(a, b)$ , there is a $c$ in $(a, b)$ with $f'(c) = \dfrac{f(b) - f(a)}{b - a}$ . Some tangent is parallel to the chord. The average rate of change equals an instantaneous rate. The theorem proves inequalities and shows that a function with zero derivative is constant.
1 simulation1 practice question1 Python program3 problems1 Python lab1 question set1 note
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Higher Derivatives and Mean Value Theorems: Chapter Test

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Higher Derivatives and the Mean Value Theorems

Course outline (3 lessons)

Second and higher derivatives

Rolle's theorem

The Mean Value Theorem and its uses

Test Your Learning

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