Calculus 1000A/B Lecture Notes - Lecture 2: Differentiable Function, Intermediate Value Theorem

• School: Western University
• Department: Calculus
• Course: Calculus 1000A/B
• Professor: Rasul Shafikov

Document Summary

A function will not have a derivative at a point. , if the slopes of the secant lines fail to approach a common limit. Examples: a corner, a vertical tangent, a cusp, a discontinuity: Differentiable function: a function that is differentiable at every point in its domain. Polynomials, rational functions, trigonometric functions, exponential and logarithmic functions are all differentiable. Composites of differentiable functions are differentiable; as are sums, products, integer powers and quotients. If we zoom in on a differentiable function, it will seem to straighten out and look like the tangent line to the point. Graphing calculators will find a numerical derivative by using the symmetric difference quotient to approximate. The arguments are nderiv (function, variable, specified value). *note: it is possible to fool the calculator as it will tend to not pick up when a function is not differentiable. However, according to our calculator, nderiv (abs(x), x, 0) = 0. *note: the converse of this theorem is false.