MATH 151 Engineering Mathematics I Fall, 2016, WEEK 4 JoungDong Kim Week4 Section 2.6, 2.7, 3.1 Limits at infinity, Velocity, Differentiation Section 2.6 Limits at Infinity; Horizontal Asymptotes Definition. Let f be a function defined on some interval (a, ). Then lim f(x) = L x means that the values of f(x) can be made arbitrarily close to L by taking x sufficiently large. Definition. Horizontal Asymptote The line y = L is called a horizontal asymptote of the curve y = f(x) if either lim f(x) = L or lim f(x) = L. x x 1
Limit of Power Function at Infinity If p is a positive real number, lim x ± 1 x p = 0. Finding limits at infinity for a rational function, f(x): Look for the highest degree of x: 1. If it is in the denominator, then lim f(x) = 0. x ± 2. If it is in the numberator, then lim f(x) = ±. x ± 3. If the degree of the polynomial in the numberator and denominator is the same then lim x ± f(x) = ratio of the leading coefficients. Ex1) Find the limits: a) lim x x b) lim x x c) lim x (x x 2 ) d) lim x (x x3 ) 1 e) lim x x 1 f) lim x x 1 g) lim x x 4 h) lim x 7x 3 +4x 2x 3 x 2 +3 2
i) lim t t 4 t 2 +1 t 5 +t 3 t x 4 +2x+3 j) lim x x(x 2 1) k) lim x 1+4x 2 4+x l) lim x 1+4x 2 4+x m) lim x x2 +4x 4x+1 n) lim x ( x 2 +3x+1 x) o) lim x (x+ x 2 +2x) 3
Finding the Vertical Asymptote and Horizontal Asymptote. 1. Vertical asymptote: undefined point but if it could be cancelled, it is not vertical asymptote but hole. 2. Horizontal asymptote: use infinite limit x and x. Ex2) Find the equation of all vertical and horizontal asymptotes. a) f(x) = x+3 x 2 +7x+12 b) f(x) = x x2 +1 4
Section 2.7 Tangents, Velocities, and other rate of change If a curve C has equation y = f(x) and we want to find the tangent to C at the point P(a,f(a)), then we consider a nearby point Q(x,f(x)), where x a, and compute the slope of the secant line PQ: m PQ = f(x) f(a) x a Then we let Q approach P along the curve C by letting x approach a. If m PQ approaches a number m, then we define the tangent t to be the line through P with slope m. (This amounts to saying that the tangent line is the limiting position of the secant line PQ as Q approaches P.) Definition. Tangent Line The tangent line to the curve y = f(x) at the point P(a,f(a)) is the line through P with slope f(x) f(a) m = lim x a x a or let x = a+h, then provided that this limit exists. m = lim h 0 f(a+h) f(a) h 5
Ex3) Find the slope of the tangent line to the graph of f(x) = x 2 +2x at the point (1,3). Ex4) Find the equation of tangent line to the graph of f(x) = 2x+5 at the point is x = 2. 6
Ex5) Find the equation of the tangent line to the graph of f(x) = 1 x+2 at x = 3. 7
Velocities: Linear motion If f(t) is the position of an object at time t, then 1. The Average Velocity of the object from t = a to t = b is f t = f(b) f(a) b a 2. The Instantaneous Velocity of the object at time t = a is v(a) = lim h 0 f(a+h) f(a) h Ex6) The position (in meters) of an object moving in a straight path is given by s(t) = t 2 8t+18, where t is measured in seconds. a) Find the average velocity over the time interval [3,4]. b) Find the instantaneous velocity at time t = 3. 8
Other rate of change Let f(x) be a function 1. The Average rate of change of f(x) from x = a to x = b is f(b) f(a) b a 2. The Instantaneous Rate of change of f(x) at x = a is f(a+h) f(a) lim h 0 h Ex7) If f(x) = x, and a) the average rate of change of f(x) from x = 4 to x = 9. b) the instantaneous rate of change of f(x) at x = 4. 9
Ex8) The population P (in thousands) of a city from 1990 to 1996 is given in the following table. Year Population ( 1000) 1990 105 1991 110 1992 117 1993 126 1994 137 1995 150 1996 164 a) Find the average rate of growth from 1992 to 1994 b) Estimate the instantaneous rate of growth in 1992 by measuring the slope of a tangent. 10
Ex9) The limit below represents the instantaneous rate of change of some function f at some number a. State such an f and a: (2+h) 5 32 lim h 0 h 11
Chapter 3. Derivatives Section 3.1 Derivatives Definition. Derivative The Derivative of a function f at a number a, denoted by f (a), is or if this limit exists. f (a) = lim h 0 f(a+h) f(a) h f (a) = lim x a f(x) f(a) x a 12
Ex10) Find the derivative of the following functions at the number a given. (use definition) a) f(x) = x 2 8x+9, a = 2 b) f(x) = x x+1, a = 2 13
Interpretations of the Derivatives Geometrically the tangent line to y = f(x) at (a,f(a)) is the line through (a,f(a)) whose slope is equal to f (a), the derivative of f at a. 1. The slope of the tangent line to the graph of f(x) at x = a. 2. The instantaneous rate of change of f(x) at x = a. 3. The instantaneous velocity at x = a. All use f(a+h) f(a) lim h 0 h Ex11) Recall the surface area of a sphere is given by A = 4πr 2. Find the average rate of change of the area from r = 1 to r = 2. Find the instantaneous rate of change of the area at r = 1. 14
Definition. Differentiable A function f is differentiable at a if f (a) exists. Theorem. If f is differentiable at a, then f is continuous at a. How can a function FAIL to be differentiable 1. If the graph of a function f has a corner or kink in it, then the graph of f has no tangent at that point and f is not differentiable there. 2. If f is not continuous at a, then f is not differentiable at a. 3. The curve has a vertical tangent line when x = a, f is not differentiable at a. 15
Ex12) The graph of f is given. State, with reasons, the numbers at which f is not differentiable. Ex13) Where is f(x) = x 2 4 not differentiable. Ex14) Where is f(x) = x+1 x+1 not differentiable. 16
Ex15) Given the graph of f(x) below, sketch the graph of the derivative. a) b) 17
Definition. Derivative of function The derivative of f is defined as f (x) = lim h 0 f(x+h) f(x) h Ex16) Find the derivative of the following functions as well as the domain of the derivative. a) f(x) = x 2 8x+9 b) f(x) = 3x+1 x 2 18
c) f(x) = 4 x 19
Ex17) The limit below represent the derivative of some function f(x) at some number a. Identify f(x) and a for each limit. a) lim h 0 (2+h) 5 32 h b) lim x 3π cosx+1 x 3π 20