Calculus on Demand at Dartmouth College Lecture 7 | Index | Lecture 9
Lecture 8


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In this lecture we define the concept of continuity in terms of limits. We also study the notion of the limit of tangent lines at a point, and the limit of their corresponding slopes at the point.


Quick Question

This is a graph of what function?


Answer

Outline

Outlines for
Continuity
Tangent Lines and their Slopes

Textbook

Continuity
Tangent Lines and their Slopes

Today's Homework

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Quiz

Continuity Quiz
Tangent Lines and their Slopes Quiz

Examples

  • Click to see the exampleFind the limit of |x| as x approaches 0 from the left and from the right. Is |x| continuous at 0?
  • Click to see the exampleConsider function f(x) that returns the biggest integer smaller than or equal to x. What can you say about the continuity of f(x)?
  • Click to see the exampleA hiker sets out at dawn to walk over a mountain and down the other side to a lake, a distance of x kilometers, and he reaches his destination exactly 12 hours later. He camps, then leaves at the same time the next morning and travels the same trail back, arriving at his starting place 12 hours later. Use the Intermediate Value Theorem to show that at some specific time, he was at the same place on his hike on both days.
  • Click to see the exampleCan you find a tangent line to f(x) = |x| at x = 0?
  • Click to see the exampleA practice ski jump hill follows the shape of a given curve. Come up with a formula for the angle the skier's skis make with the horizontal, and find how far from the top of the jump he is when this angle is the greatest.
  • Click to see the exampleA potted cactus is thrown upward with a velocity of 40 feet per second. Its height in feet at time t is given by the formula h(t) = 40t – 16t2. Find its velocity 2 seconds after it is released.

Applets

  • Click to see the appletContinuity of Functions
  • Click to see the appletSecant and Tangent Lines

Videos

  • click to see the videoLeft and right continuous defined
  • click to see the videoDefine slope of tangent line as limit of slopes of secant lines
  • click to see the videoFind slope of tangent line to y = x2 at (1,1)
  • click to see the videoFind slope of tangent line to y = √x at x = 4
  • click to see the videoFind the equation for the tangent line of 4x2 + 10x + 5 at the point (2,1) using the limit definition of the derivative
Describe continuity of:
  • click to see the videof(x) = 3x for x < 1 f(x) = 4x for x ≥ 1
  • click to see the videof(x) = 2x for x < 0, f(x) = x3 for x ≥ 0

Lecture 7 | Index | Lecture 9