Calculus on Demand at Dartmouth College Lecture 11 | Index | Lecture 13 Lecture 12

## Resources

Math 3 Course Syllabus
Practice Exams

# Contents

In this lecture we study something called the "Mean Value Theorem." This theorem allows us to take an "at-a-point" concept like the derivative and use it to study the global behavior of a function. We will also learn how to differentiate a function for which we do not have an explicit formula.

### Quick Question

Suppose Area I (above the blue line and below the parabola) equals Area II (above the parabola and below the blue line). What is the total area enclosed by the parabola?

### Outline

Outlines for
The Mean Value Theorem
Implicit Differentiation

### Textbook

The Mean Value Theorem
Implicit Differentiation

### Quiz

The Mean Value Theorem Quiz
Implicit Differentiation Quiz

### Examples

• At 7 p.m., a car is traveling at 50 miles per hour. Ten minutes later, the car has slowed to 30 miles per hour. Show that at some time between 7 and 7:10 the car's acceleration is exactly 120, in units of miles per hours squared.
• At a particular horse race, two horses start at the same time, and finish in a tie. Show that at some time during the race, the horses were running at the same speed.
• Suppose f is a differentiable function such that f(1) = 20, f'(x) ≥ 3, 1 ≥ x ≥ 6. What is the smallest possible value for f(6)?
• Find y' by implicit differentiation, where xy = cot(xy).
• Find the tangent line to an ellipse at a given point.
• An interesting curve first studied by Nicomedes around 200 B.C. is the conchoid, which has the equation x2y2 = (x + 1)2 (4 – x2). Use implicit differentiation to find a tangent line to this curve at the point (–1, 0).

### Applets

• Mean Value Theorem

### Videos

• Tangent is positive where function is increasing, negative where function is decreasing
• 3x2 + 12x
• x3 + 17 – 12x
• The second derivative is the derivative of the derivative
• sin(x2 + 2)
• sin(2x)

Lecture 11 | Index | Lecture 13