1. Definition of Derivative
Calculus overview 3 main concepts Limits Derivatives Integrals
What is the derivative? The derivative is the slope of the tangent line at a point. Therefore if you have a graph, you can estimate the derivative by looking at the slope of the graph at that point
Example 1 Estimate the derivative at the points marked on the graph below.
Slope of secant line vs. tangent line As h shrinks to zero, slope of secant line approaches slope of tangent line http://faculty.wlc.edu/buelow/calc/nt2-1.html Slope of tangent line =
Derivative The formula for slope of a tangent line (with the limit) is called the derivative Can also be written This will give a numeric value
Secant vs. tangent Secant Line Tangent Line Uses 2 points (a, f(a)) and (b,f(b)) Uses 1 point (c, f(c)) The following are equivalent: Slope of secant line between [a,b] Slope of tangent line at x = c Average rate of change on [a,b] Instantaneous rate of change at x=c f’(c) Derivative at x = c dy/dx
Example 2 For f(x) = 4x2 , use the definition of the derivative to find the derivative function f’(x), then find the equation of the tangent line to f(x) at x=2.
Example 3 For g(t) = t/(t+1), use the definition of the derivative to find the derivative function g’(t), then find the equation of the normal line to g(t) at t=1.
Example 4 Find the derivative using the alternative from of the definition of f(x)= x2 – 3x at x=1
Differentiability A function f(x) is differentiable at x=c if and only if L is a finite value This basically says that for a function to be differentiable at a point, the graph must be continuous but also must connect in a way that the slopes merge into each other smoothly.
Example 5 Discuss the differentiability of at x = 2
Example 6 Discuss the differentiability of at x = 0
Differentiability Places a function is not differentiable Anywhere there is a discontinuity Corner or cusp Vertical tangent
Local Linearity A function that is differentiable closely resembles its own tangent line when viewed very closely In other words, when zoomed in on a few times, a curve will look like a straight line if it is differentiable.
Example 7
Continuity vs. Differentiability If f(x) is differentiable at x = c, then f is continuous at x = c. Be careful! The converse is not true! Being continuous does not imply differentiability (think about the absolute value function)
Piecewise function You must always be careful about piecewise functions Example 8 For the piecewise function find a piecewise derivative function f’(x) by using the definition of the derivative. Then evaluate the following What can you say about the differentiability based on your results?