alittlebitofcalculus-pdf-january2011-111112001007-phpapp01

The first par t of the study of calculus deals with

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 25 m/sec for t ≥ 10 seconds is constant. Limit v(t) − v(10) v(t) − v(10) ≠ right derivative t → 10+ left derivative Limit0− right t→1 t − 10 t − 10 we say : v(t) is not differentiable at t = 10 secs. differentiable Even though v(t) is not differentiable at t = 10 secs, we can still compute the left differentiable left derivative and right derivative and interpret them according to the situation. deriv tiv right deriv tiv Exer ercise1: Exercise1: Given f ’(x) = AVERAGE RATE OF CHANGE of f(x), what is f(x) ? a) ex b) |x| c) constant d) x e) ax + b 90 Example 4: Is the function y(t), that describes the height of a bouncing ball, bouncing differentiable at instants t0, t1, t2, t3, . . . ? y(t) SLOPE of tangent at t1 from the left is negative left negative h e i g h t t0 SLOPE of tangent at t1 from the right is postive right postive n t1 t2 t3 t ertical At t1 we have two instantaneous ver tical speeds : the speed on impact at t1 while instantaneous ver descending and the speed after impact at t1 while rising. We know the duration of rising flight T = 2u0. sinθ/ g (see page 75). To find the speed on impact at t1 while descending we substitute T = 0 and t = T0 in y0’(t) = u0. sinθ0 − g(t − t0) to get y0’(t− ) = −u0. sinθ0 . 1 After impact at t1 we know this speed on rising is y1’(t+ ) = +u1.sinθ1. Alternatively, rising 1 we may substitute t = t1 in y1’(t) = u1.sinθ1 − g(t − t1) to get the same result. Geometrically Geometrically, we can see that we have two tangents at t1. left derivative y0’(t− ) = −u0.sinθ0 ≠ 1 y1’(t+ ) = +u1.sinθ1 right derivative right 1 we say : y(t) is not differentiable at t = t1 secs. differentiable Even though y(t) is not differentiable at t = t0, t1, t2, t3, . . . , we can still differentiable compute the left derivative and right derivative and interpret them according to left deriv tiv right deriv tiv the situation. 91 Example 5 : is f(x) = + √x differentiable at x = 0 ? differentiable f(x) f(x) = +√ x (0, 0) x We know that f(x) =...
View Full Document

Ask a homework question - tutors are online