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
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
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
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
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
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
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
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
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
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
u) lim ( x 2 +3x+1 x) v) lim x (x+ x 2 +2x) 11
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
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
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 x2 +1 14
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
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
Ex.14) Find the equation of the tangent line to the graph of f(x) = 1 x+2 at x = 3. 17
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
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
Ex.17) 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. 20