Derivative of a Function (x+δx,f(x+δx)) f ' (x) = (x,f(x)) provided the limit exists Can be interpreted as the slope of the tangent line to the curve at any point (x, f(x)) on the curve. This generalizes from the Derivative at a specific point (c,f(c)) to any point on the curve (x,f(x)) Notation: f '(x) dy G. Battaly, WCC, Class Notes Homework 1. The Constant Rule Rules of Differentiation d [ c ] = 0 2. The Power Rule d[ x n ] = nx n 1 3. The Constant Multiple Rule d [ c f(x) ] = c f '(x) 4. The Sum & Difference Rule 5. Trig Functions d [ f(x)± g(x) ] = f '(x) ± g '(x) d [ sin x ] = cos x d [ cos x ] = sin x G. Battaly 2017 1
co functions d(sin x) = cos x d(tan x) = sec 2 x d(cos x) = sin x d(cot x) = csc 2 x d(sec x) = sec x tan x d(csc x) = csc x cot x Homework Chain Rule If: 1) y = f(u) is a differentiable function of u, and 2) u=g(x) is a differentiable function of x, then y = f(g(x)) is a differentiable function of x, and or Homework G. Battaly 2017 2
Quick Review of CALC 1 co functions d(sin u) = cos u du d(tan u) = sec 2 u du d(sec x)=secu tanu du d(cos u) = sin u du d(cot u) = csc 2 u du d(csc x)= cscu cotu du derivative practice Homework More Rules of Differentiation where u = f(x) and v = g(x) 1. The Product Rule d [ u v ] = u v' + v u' 2. The Quotient Rule d[ u / v ] = v u' u v' v 2 3. The Natural Logarithm 4. The Exponential Function d [ ln u ] d [ e u ] = u ' u = e u u ' 5. Inverse Trig Functions d [ arcsin u ] = u ' 1 u 2 d [ arccos u ] = u ' 1 u 2 d [ arctan u ] = u ' 1 + u 2 G. Battaly 2017 3
NOT calculus Average Rate of Change on Interval Slope of Secant Line eg: total distance total time vs CALCULUS Instantaneous Rate of Change Slope of Tangent Line eg: instantaneous velocity Curve Sketching Goal: Sketch graphs of functions using 1. Domain and range 2. Asymptotes 3. Relative Extrema 4. Concavity and Inflection Points. geogebra Other Applications 1. Max / Min: Optimization Problems 2. Related Rates Derivative Sketches Homework G. Battaly 2017 4
Definition of Area of a Region in a Plane 1 2 Let f be continuous and non negative on [a,b]. The area of the region bounded by the graph of f, the x axis, and the vertical lines x = a and x = b is: x i 1 < c i < x i Δx = b a n as n >, Δx > 0 Homework on Web Definition of Riemann Sum Let f be defined on [a,b], and let Δ be a partition of [a,b], given by a = x 0 < x 1 < x 2 <... < x n 1 < x n =b where Δ x i is the width of the ith sub interval If c i is any point on the ith sub interval, then the sum Ʃ f(c i ) Δ x i, x i 1 < c i < x i is called a Riemann Sum of f for the partition Δ Δ x i not all equal norm of the partition Δ = width of longest subinterval G. Battaly 2017 5
Definition of Definite Integral If f is defined on [a,b], and the following limit exists lim Ʃ f(c i ) Δ x i Δ >0 Then f is integrable on [a,b] and the limit is denoted as: lim Ʃ f(c i ) Δ x i Δ >0 This is called the Definite Integral of f from a to b Definition, Antiderivative: A function F is an antiderivative of f on an interval I if F ' (x) = f(x) for all x in I. A Differential Equation in x and y is an equation that involves x, y, and the derivative of y. eg: dy = 5x 4 6x 2 Homework on Web G. Battaly 2017 6
Fundamental Theorem of Calculus (FTC) If : 1. a function f is continuous on [a, b] and then: 2. F is an antiderivative of f on the interval, b f(x) = F(b) F(a) a The integral of f from a to b is the difference: (antiderivative of f evaluated at x=b) (antiderivative of f evaluated at x=a.) Homework Part 2 Fundamental Theorum of Calculus If f is continuous on an open interval, I, containing a, then, for every x on the interval: d x f(t)dt = f(x) a Homework Part 2 G. Battaly 2017 7
Fundamental Theorum of Calculus If f is continuous on an open interval, I, containing a, and u = f(x) then, for every x on the interval: d u f(t)dt = f(u) u' a Homework Part 2 Rules of: Differentiation Integration d( c ) = 0 0 = c d(kx) = k k = kx + c d(sinx) = cos x cosx = sinx + c d(cosx) = -sin x sinx = -cosx + c Class Notes: Prof. G. Battaly, Westchester Community College, NY Homework on Web sinx = - cosx + c G. Battaly 2017 8