MATH 151 Engineering Mathematics I

Transcription

1 MATH 151 Engineering Mathematics I Spring 2018, WEEK 3 JoungDong Kim Week 3 Section 2.5, 2.6, 2.7, Continuity, Limits at Infinity; Horizontal Asymptotes, Derivatives and Rates of Change. Section 2.5 Continuity Definition A function f is continuous at a number a if limf(x) = f(a) x a Note Definition implicitly requires three things if f is continuous at a; 1. f(a) is defined (that is, a is in the domain of f and has a function value at a) ( ) 2. lim x a f(x) exists. 3. lim x a f(x) = f(a) lim x a +f(x) = lim x a f(x) 1

2 In each case the graph cannot be drawn without lifting the pen from the paper, because a hole or break or jump occurs in the graph. Definition. The kind of discontinuity illustrated first and three is called removable becuase we could remove the discontinuity by redefining f at 2. The discontinuity in second figure is called an infinite discontinuity. The discontinuities in the last figure are called jump discontinuities because the function jumps from one value to another. Definition. A function f is continuous from the right at x = a if lim x a +f(x) = f(a) and if f is continuous from the left at x = a if lim x a f(x) = f(a) 2

3 Theorem. (a) Any polynomial is continuous everywhere; that is, it is continuous on R = (, ). (b) Any rational function is continuous wherever it is defined; that is, it is continuous on its domain. Ex.1) Show that f(x) = x 2 3x+2 is continuous at x = 2. Ex.2) From the accompaning figure, state the numbers at which f is discontinuous. For each of the numbers stated, state whether f is continuous from the right, or from the left, or neither. 3

4 Ex.3) Explain why the following functions are not continuous at the indicated values of x. a) f(x) = 1 (1 x) 2, x = 1 2x+1 if x 0 b) f(x) = 3x if x > 0, x = 0 x 2 2x 8 if x 4 c) f(x) = x 4 3 if x = 4, x = 4 4

5 x 2 c 2 if x < 4 Ex.4) If g(x) = cx+20 if x 4. For what value(s) of c make(s) g(x) continuous? 5

6 Intermediate Value Theorem Suppose that f is continuous on the closed interval [a,b] and let N be any number strictly between f(a) and f(b). Then there exists a number c in (a,b) such that f(c) = N. Ex.5) Show there is a root of the equation 4x 3 6x 2 +3x 2 = 0 between 1 and 2. Ex.6) If g(x) = x 5 2x 3 +x 2 +2, show there a number c so that g(c) = 1 in [ 2, 1]. 6

7 Section 2.6 Limits at Infinity; Horizontal Asymptotes Definition. Let f be a function defined on some interval (a, ). Then lim f(x) = L 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 Ex.7) Find limit. a) lim arctanx b) lim x arctanx 7

8 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 f(x) = ratio of the leading coefficients. lim x ± Ex.8) Find the limits: a) lim x b) lim x x c) lim (x x 2 ) d) lim x (x x3 ) 8

9 1 e) lim x 1 f) lim x x 1 g) lim x 4 h) lim 7x 3 +4x 2x 3 x 2 +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 1+4x 2 4+x l) lim x 1+4x 2 4+x m) lim x x2 +4x 4x+1 9

10 n) lim (0.3) x o) lim x (0.3) x p) lim x 2 + ( ) x 1 2 x 4 q) lim x 2 ( ) x 1 2 x 4 r) lim 2 x 1 e x 1 s) lim 2 x 1 +e x 1 t) lim e x e 3x e 3x +e 3x 10

11 u) lim ( x 2 +3x+1 x) v) lim x (x+ x 2 +2x) 11

12 Theorem. If a > 1, the function f(x) = log a x is one-to-one, continuous, increasing function with domain (0, ) and range R. If x, y > 0, then 1. log a (xy) = log a x+log a y ( ) x 2. log a = log y a x log a y 3. log a (x y ) = ylog a x 4. log a a = 1 5. a log a x = x 6. change of base: log a b = log cb log c a or log ab = 1 log b a 7. log a 1 = 0 Ex.9) Find the limit; a) lim (ln(3x 2 2x+5) ln(2x 2 +4x)) 12

13 b) lim (ln(3x 2 ) ln(6x 4 3x+1)) Ex.10) Find the limits; a) lim arctan(e x ) b) lim arctan(lnx) c) lim x 0 +arctan(lnx) 13

14 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. Ex.11) Find the equation of all vertical and horizontal asymptotes. a) f(x) = x+3 x 2 +7x+12 b) f(x) = x x

15 Section 2.7 Derivatives and Rates 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 m = lim x a f(x) f(a) x a or let x = a+h, then provided that this limit exists. m = lim h 0 f(a+h) f(a) h 15

16 Ex.12) Find the slope of the tangent line to the graph of f(x) = x 2 +2x at the point (1,3). Ex.13) Find the equation of tangent line to the graph of f(x) = 2x+5 at the point is x = 2. 16

17 Ex.14) Find the equation of the tangent line to the graph of f(x) = 1 x+2 at x = 3. 17

18 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 Velocity (or Instantaneous Velocity) of the object at time t = a is v(a) = lim h 0 f(a+h) f(a) h Ex.15) 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. 18

19 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 Ex.16) If f(x) = x, a) Find the average rate of change of f(x) from x = 4 to x = 9. b) Find the instantaneous rate of change of f(x) at x = 4. 19

20 Ex.17) 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. 20