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


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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?


Answer

Outline

Outlines for
The Mean Value Theorem
Implicit Differentiation

Textbook

The Mean Value Theorem
Implicit Differentiation

Today's Homework

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Quiz

The Mean Value Theorem Quiz
Implicit Differentiation Quiz

Examples

  • Click to see the exampleAt 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.
  • Click to see the exampleAt 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.
  • Click to see the exampleSuppose 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)?
  • Click to see the exampleFind y' by implicit differentiation, where xy = cot(xy).
  • Click to see the exampleFind the tangent line to an ellipse at a given point.
  • Click to see the exampleAn 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

  • Click to see the appletMean Value Theorem

Videos

  • click to see the videoTangent is positive where function is increasing, negative where function is decreasing
  • click to see the video3x2 + 12x
  • click to see the videox3 + 17 – 12x
  • click to see the videoThe second derivative is the derivative of the derivative
  • click to see the videosin(x2 + 2)
  • click to see the videosin(2x)

Lecture 11 | Index | Lecture 13