MATH 151 Engineering Mathematics I

Transcription

1 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

2 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

3 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

4 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

5 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

6 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

7 Ex5) Find the equation of the tangent line to the graph of f(x) = 1 x+2 at x = 3. 7

8 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

9 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

10 Ex8) The population P (in thousands) of a city from 1990 to 1996 is given in the following table. Year Population ( 1000) 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

11 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

12 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

13 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

14 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

15 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

16 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

17 Ex15) Given the graph of f(x) below, sketch the graph of the derivative. a) b) 17

18 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

19 c) f(x) = 4 x 19

20 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