The left-hand and right-hand derivatives off at a are defined by f ' ( a ) = lim h → 0 − f ( a + h ) − f ( a ) a and f ' + ( a ) = lim h → 0 + f ( a + h ) − f ( a ) h if these limits exist. Then f' ( a ) exists if and only if these one-sided derivatives exist and are equal. (a) Find f ' − ( 4 ) and f ' + ( 4 ) for the function f ( x ) = { 0 if x ≤ 0 5 − x if 0 < x < 4 1 5 − x if x ≥ 4 (b) Sketch the graph of f (c) Where is f discontinuous? (d) Where is f not differentiable?
The left-hand and right-hand derivatives off at a are defined by f ' ( a ) = lim h → 0 − f ( a + h ) − f ( a ) a and f ' + ( a ) = lim h → 0 + f ( a + h ) − f ( a ) h if these limits exist. Then f' ( a ) exists if and only if these one-sided derivatives exist and are equal. (a) Find f ' − ( 4 ) and f ' + ( 4 ) for the function f ( x ) = { 0 if x ≤ 0 5 − x if 0 < x < 4 1 5 − x if x ≥ 4 (b) Sketch the graph of f (c) Where is f discontinuous? (d) Where is f not differentiable?
Solution Summary: The author explains how to calculate the left-hand derivative of f at x=a.
a
->
f(x) = f(x) = [x] show that whether f is continuous function or not(by using theorem)
Muslim_maths
Use Green's Theorem to evaluate F. dr, where
F = (√+4y, 2x + √√)
and C consists of the arc of the curve y = 4x - x² from (0,0) to (4,0) and the line segment from (4,0) to
(0,0).
Evaluate
F. dr where F(x, y, z) = (2yz cos(xyz), 2xzcos(xyz), 2xy cos(xyz)) and C is the line
π 1
1
segment starting at the point (8,
'
and ending at the point (3,
2
3'6
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Differential Equation | MIT 18.01SC Single Variable Calculus, Fall 2010; Author: MIT OpenCourseWare;https://www.youtube.com/watch?v=HaOHUfymsuk;License: Standard YouTube License, CC-BY