Math 103, week 4-5 notes
Math 103, week 4-5 notes MATH 103 001
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This 4 page Class Notes was uploaded by Cambria Revsine on Monday February 22, 2016. The Class Notes belongs to MATH 103 001 at University of Pennsylvania taught by William Simmons in Spring 2016. Since its upload, it has received 16 views. For similar materials see Intermediate Algebra Part III in Mathematics (M) at University of Pennsylvania.
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Date Created: 02/22/16
Math 103—Week 45 Notes—3.23.5 3.2: Derivatives: The derivative of a function f(x) at any point x: f (x)=lim f(x+h )− f (x) h→0 h ' f(z − f (x) f (x)=lz→ x z−x *z = x + h, or the second point Derivative notations: ' ' dy df d f (x =y = = = f(x =D (f)(x)=D fx(x) dx dx dx One can graph a derivative by estimating the slope at several points on the function, plotting the points as (x , f(x)) on a second graph, and connecting the dots with a smooth curve Differentiable: when a function has a derivative at every point, or at a specific point a function is differentiable on an open interval if it has a derivative at every point on the interval a function is differentiable on a closed interval [a,b] if it has a derivative on the interior (a,b) and a righthand derivative at a and lefthand derivative at b Where is a function not differentiable at a point? *Whenever the right and lefthand derivatives are not equal 1. A corner (like f x = |x| at x=0) ¿ x∨¿ 2. A cusp (like f x )= ¿ at x=0) 3. A vertical tangent (where the slope is vertical; undefined) 4. A discontinuity (where there is a jump or point missing) 1 5. Rapid oscillation of the slope (like at f(x)=sin? (x) at x=0) **If f is differentiable at a point, it is continuous at the point the converse is not necessarily true 3.3: Differentiation Rules: Derivative of a Constant Function: d If f x =c then dx(c)=0 Power Rule: d (xn=nx n−1 dx Derivative Constant Multiple Rule: If u is a differentiable function ox and c is a constant, then d (cu)=c du dx dx d du dx cu =c dx Sum Rule: If u and v are differentiable functions ofx then their sum u+v is differentiable, d du dv and (u+v = + dx dx dx Ex: Find the points where the curve y=x −2x +22 has horizontal tangents. Horizontal tangents occur where the slope of an equation equals 0: d ( x −2x +2¿=4x −4x 3 dx 4 x −4x=0 2 4 x(x −1 )=0 x=0,1,−1 Plug the xvalues back in to original equation: 0,2 ,(1,1),(−1,1) Derivative of e : d x x dx (e =e Product Rule: If u and v are differentiable atx then their product uv is differentiable, and d dv du uv =u +v dx dx dx “First times derivative of the second plus second times derivative of the first” Quotient Rule: u If u and v are differentiable atx and if v(x)≠0 , then the quotient is v differentiable, and du dv d u v dx −u dx = 2 dx v) v “Bottom times derivative of the top minus top times derivative of the bottom all over bottom squared” Second derivative: derivative of the first derivative Second derivative notations: 2 f'(x = d y = d dy = dy' =y =D' 2(f)x =D f (x) d x2 dx dx dx x th **You can keep taking the derivative any subsequent number of times, to the n derivative 3.4: Rates of Change: Instantaneous rate of change of f with respect to x a0 x equals the derivativ0 of x . ' f(x0+h ) f (x0) f ( 0lim h→0 h *Instantaneous rates are limits of average rates ** “rate of change” usually means “instantaneous rate of change” Position: s= f (t) Velocity: derivative of position with respect to time t ds f(t+∆t )− f (t) v(t= = lim dt ∆t →0 ∆t *Positive velocity means it is moving forward, negative velocity means it is moving backward Speed: absolute value of velocity | | =¿ ds∨¿ dt Acceleration: derivate of velocity with respect to time: If a object’s position at time t is s=f(t) then its acceleration at time t is dv d s a(t= = 2 dt dt *rate of change of an object’s velocity (how quickly an object increases or decreases speed) Jerk: derivative of acceleration with respect to time da d s j(t)= = 3 dt dt *sudden change in acceleration Derivatives in Economics: Marginal cost of production: derivative of cost of production with respect to x (the number of units produced) AKA average cost of each additional unit produced dc dx Sensitive: a term for when a small change in x produces a large change in f(x) When x is small, its change produces a larger change in y In the derivative graph, a higher value means it is more sensitive at that point 3.5: Trig Derivatives: Derivative of sin x: cos x Derivative of cos x: sin x 2 Derivative of tan x: sec x Derivative of csc x: csc x cot x Derivative of sec x: sec x tan x Derivative of cot x: csc x