Lecture Notes for Math 1000 Dr. Xiang-Sheng Wang Memorial University of Newfoundland Office: HH-2016, Phone: 864-4321 Office hours: 13:00-15:00 Wednesday, 12:00-13:00 Friday Email: xswang@mun.ca Course website: http://www.ucs.mun.ca/~xiangshengw/1000.html Lecture Notes for Math 1000 First Previous Next Last 1
Rates of change The average rate of change of a function y = f(x) over an interval [x 0, x 1 ] is defined as f(x 1 ) f(x 0 ) x 1 x 0. The instantaneous rate of change of a function y = f(x) at x = x 0 is defined as f(x 1 ) f(x 0 ) f(x 0 + h) f(x 0 ) lim = lim. x 1 x 0 x 1 x 0 h 0 h The average rate of change is equal to the slope of the secant line through (x 0, f(x 0 )) and (x 1, f(x 1 )). The instantaneous rate of change is equal to the slope of the tangent line at x = x 0. Lecture Notes for Math 1000 First Previous Next Last 2
The definition of the derivative The derivative of a function f(x) at x = a is the limit of the difference quotient (if exists): f (a) = lim x a f(x) f(a) x a f(a + h) f(a) = lim. h 0 h If f (a) exists, then we say that f is differentiable at x = a. Theorems. 1. If f(x) = mx + b, then f (a) = m for all a. 2. If f(x) is differentiable at x = a, then f(x) is continuous at x = a. Remark. If f is continuous at x = a. It is NOT necessarily that f is differentiable at x = a. (Counter example: f(x) = x and a = 0.) We say y = f(x) is differentiable in (a, b) if f (x) exists for all x in (a, b). In this case, we view y = f (x) as a function defined on (a, b). Lecture Notes for Math 1000 First Previous Next Last 3
Notations Let y = f(x), then its derivative function is denoted by For example, if y = f(x) = x 1, then y = f (x) = dy = df = d (f(x)). y = f (x) = dy = df = d (x 1 ) = x 2. If y = f(x) is differentiable at x = a, then we write y (a) = f (a) = dy = df x=a. x=a For example, if y = f(x) = x 1 and a = 1, then y (1) = f (1) = dy = df x=1 = ( x 2) x=1 = 1. x=1 Lecture Notes for Math 1000 First Previous Next Last 4
Derivatives of algebraic functions and exponential functions Two formulas: (x a ) = ax a 1 and (e x ) = e x. The tangent line to y = f(x) at x = a has the slope f (a) and the equation in point-slope form: y f(a) = f (a)(x a). Example: The tangent line of the curve y = x 3 at x = 1 is given by y 1 = 3(x 1). Example: The tangent line of the curve y = e x at x = 0 is given by y 1 = x. Lecture Notes for Math 1000 First Previous Next Last 5
The properties of derivatives Let f(x) and g(x) be differentiable functions. The Sum Rule: The Constant Multiple Rule: The Product Rule: (f ± g) = f ± g. (cf) = cf. (fg) = f g + fg. The Quotient Rule: ( f g ) = f g fg g 2. Remark:. (fg) f g and ( ) f f g g Lecture Notes for Math 1000 First Previous Next Last 6
Two basic formulas: Derivatives of trigonometric functions (sin x) = cos x and (cos x) = sin x Express everything in terms of sin x and cos x and then apply derivative rules: ( ) d d sin x (tan x) = = 1 cos x cos 2 x = sec2 x. d d (cot x) = d d (sec x) = d d (csc x) = ( cos x ) = 1 sin x sin 2 x = csc2 x. ( ) 1 = sin x cos x cos 2 = sec x tan x. x ( ) 1 = cos x sin x sin 2 = csc x cot x. x Lecture Notes for Math 1000 First Previous Next Last 7
Derivatives of hyperbolic functions Two basic formulas: (sinhx) = coshx and (coshx) = sinhx Express everything in terms of sin x and cos x and then apply derivative rules: (tanhx) = (coth x) = (sechx) = (cschx) = ( ) sinhx = (sinhx) (coshx) (sinhx)(coshx) coshx cosh 2 x ( ) coshx = (coshx) (sinhx) (coshx)(sinhx) sinhx sinh 2 x ( ) 1 = (1) (coshx) (1)(coshx) coshx cosh 2 x ( ) 1 = (1) (sinhx) (1)(sinhx) sinhx sinh 2 x = 1 cosh 2 x = sech2 x = 1 sinh 2 x = csch2 x = sinhx cosh 2 x = tanhxsechx = coshx sinh 2 x = coth xcschx Lecture Notes for Math 1000 First Previous Next Last 8
Higher derivatives Higher derivatives: let y = f(x), then y = f (x) = dy = df = d y = f (x) = d ( ) dy (f(x)) = d ( ) dy = d2 y 2 = d2 f 2 = d y = f (x) = y (3) = f (3) (x) = d3 y 3 = d3 f 3 = d ( d ( ) d (f(x)) )) ( d (f(x)) Given the displacement function s(t), the velocity function is given by v(t) = s (t), and the acceleration function is given by a(t) = s (t) = v (t). Lecture Notes for Math 1000 First Previous Next Last 9
The chain rule If u = g(x) is differentiable at x = x 0 and y = f(u) is differentiable at u = u 0 = g(x 0 ), then y = f(g(x)) is differentiable at x = x 0 and dy = dy x=x0 du u=u0 =g(x 0 ) du x=x0 We can also write dy = dy du du or d f(g(x)) = f (g(x))g (x) or d f(u) = f (u) du Lecture Notes for Math 1000 First Previous Next Last 10
If g(x) is differentiable, then The chain rule (special applications) d (g(x))a = a(g(x)) a 1 g (x) If f(u) is differentiable, then and d eg(x) = g (x)e g(x) Especially, if f(u) = e u, then d f(kx + b) = kf (kx + b) (e kx+b ) = ke kx+b If b > 0, then (b x ) = (ln b)b x Lecture Notes for Math 1000 First Previous Next Last 11
Given an implicit function Implicit differentiation y 4 + xy = x 3 x + 2 Take derivative with respect to x on both side of the equation d (y4 + xy) = d (x3 x + 2) Apply the chain rule and other derivative rules to obtain 4y 3dy + y + xdy = 3x2 1 Solve dy from the above equation dy = 3x2 1 y 4y 3 + x Lecture Notes for Math 1000 First Previous Next Last 12
Derivative of inverse function If y = f(x) is differentiable at x = x 0 and f (x 0 ) 0, then the inverse function x = g(y) = f 1 (y) is differentiable at y = y 0 = f(x 0 ) and Logarithmic functions g (y 0 ) = 1 f (x 0 ) = 1 f (g(y 0 )) (log x) = 1 x and (log g(x)) = g (x) g(x) Lecture Notes for Math 1000 First Previous Next Last 13
Derivatives of inverse trigonometric functions (sin 1 y) = (cos 1 y) = (tan 1 y) = (cot 1 y) = (sec 1 y) = (csc 1 y) = 1 (sin x) = 1 cos x = 1 1 sin 2 x = 1 1 y 2 1 (cos x) = 1 sin x = 1 1 cos 2 x = 1 1 y 2 1 (tan x) = cos2 x = cos 2 x cos 2 x + sin 2 x = 1 1 + tan 2 x = 1 1 + y 2 1 (cot x) = sin 2 x sin2 x = sin 2 x + cos 2 x = 1 1 + cot 2 x = 1 1 + y 2 1 (sec x) = cos2 x sin x = 1 (csc x) = sin2 x cos x = sin2 x 1 sin 2 x = cos 2 x 1 cos2 x = 1 y 2 1 (1/y) 2 = 1 y y 2 1 1 y 2 1 (1/y) 2 = 1 y y 2 1 Lecture Notes for Math 1000 First Previous Next Last 14
L Hôpital s rule Assume f (x) and g (x) exist for all x near a, and g (x) 0 for x near but not equal to a. If f(a) = g(a) = 0, then lim x a f(x) g(x) = lim x a provided that the limit on the right exists. f (x) g (x) If lim f(x) = ± and lim g(x) = ±, then L Hôpital s rule also applies. x a x a Furthermore, the limit may be taken as one-sided limit. Assume f (x) and g (x) exist for large x, and g (x) 0 for all large x. If both lim f(x) and lim g(x) are zero (or infinity), then x x lim x f(x) g(x) = lim x f (x) g (x) provided that the limit on the right exists. Similar result holds when x. Lecture Notes for Math 1000 First Previous Next Last 15