Limit and Continuity 2.1 Rate of Change and Limits (1) Average and Instantaneous Speed Example 1 FINDING AN AVERAGE SPEED A rock falls from the top of a tall cliff. What is the average speed during the first two seconds of fall ? Solution The displacement will be y (t ) 16t 2 according to the definition y y 16(2) 16(0) average speed t 2 32 ft/sec 2 2 y 2.1 Rate of Change and Limits (2, Example 2) Average and Instantaneous Speed Example 2 FINDING AN INSTANTANEOUS S PEED A rock falls from the top of a tall cliff. What is the instantaneous speed during at t 2 sec. of fall ? Solution Using the definition of average speed with t 2 and t 2 h (where h =small) y 16(2 h )2 16(2)2 AV 64 16h ft/sec t h h When different value of h AV 1 80 0.1 0.001 0.00001 65.6 64.016 64.00016 y (t ) 16t 2 y t=2 v=? 2.1 Rate of Change and Limits (3) Definition of Limit Definition Limit Let c and L be real numbers. The function f has limit L as x approaches c if, given any postive number , there is a postive number such that for all x , 0 x - c f (x ) L . lim f (x ) L We write x c y The sentence lim f (x ) L is read L x c "The limit of f of x as x approaches c equals L" c x 2.1 Rate of Change and Limits (4) Definition of Limit -2 (c) h (x ) x 1 3 3 3 2 2 2 1 1 1 0 0 0 -1 0 1 -1 2 x 2 1 (a) f (x ) x 1 -2 -1 0 1 2 -2 x 2 1 , x 1 (b) g (x ) x 1 1, x 1 -1 limf (x ) lim g (x ) limh (x ) 2 x 1 x 1 x 1 -1 0 -1 1 2 2.1 Rate of Change and Limits (5) Properties of Limit Two Basic Function Limits lim k k a constant function f (x ) k lim x c an identity function f (x ) x x c x c This can be applied to do the limits of all polynominal and rational functions. 2.1 Rate of Change and Limits (6, Theorem 1) Properties of Limit Theorem 1 Properties of Limit s If L, M , c , and k are real numbers and lim f (x ) L and lim g ( x ) M , then x c 1. Sum Rule : x c lim(f ( x ) g ( x )) L M x c The limit of the sum of two functions is the sum of their limits. 2. Difference Rule : lim(f ( x ) g ( x )) L M x c The limit of the difference of two functions is the difference of their limits. 3. Product Rule : lim(f ( x ) g ( x )) L M x The limit of a product of two functions is the product of their limits. 2.1 Rate of Change and Limits (7, Theorem 1) Properties of Limit Theorem 1 Properties of Limit s If L, M , c , and k are real numbers and lim f ( x ) L x c and lim g ( x ) M , then x c 4. Constant Multiple Rule : lim(k f ( x )) k L x The limit of a constant times a function is the constant times the limit of the function. 5. Quotient Rule : f (x ) L lim , M 0 x g ( x ) M The limit of a quotient of two functions is the quotient of their limits, provided the limit of the denominator is not zero. 2.1 Rate of Change and Limits (8, Theorem 1) Properties of Limit Theorem 1 Properties of Limits If L, M , c , and k are real numbers and lim f (x ) L x c and lim g (x ) M , then x c 6. Power Rule : If r and s are integers, s 0,then lim(f (x ))r x c provided that Lr s s Lr s is a real number. The limit of a rational power of a function is that power of the limit of the function, provided the latter is a real number. 2.1 Rate of Change and Limits (9, Example 3) Properties of Limit Example 3 USING PROPERTIES OF LIMITS Find the limit of (a) lim x 3 4x 2 3 , (b) x c x lim x c Solution (a) lim x 3 4x 2 3 lim x 3 lim 4x 2 lim 3 x c x c x c c 3 4c 2 3 (b) x lim x c 4 x 1 lim x 4 x 2 1 2 x2 5 x c lim x 2 5 x c lim x 4 lim x 2 lim1 x c x c x c lim x lim 5 2 x c x c 4 2 c c 1 c2 5 x c 4 x 2 1 x2 5 2.1 Rate of Change and Limits (10,Theorem 2) Properties of Limit Theorem 2 Polynomial and Rational Functions 1. If f (x ) an x n an 1x n 1 a 0 is any polynomial function and c is any real number. then lim f (x ) f (c ) anc n an 1c n 1 x c a0 2. If f (x ) and g (x ) are polynomials and c is any real number, then f (x ) f (c ) , x c g ( x ) g (c ) lim provided that g (c ) 0 2.1 Rate of Change and Limits (11, Example 4) Properties of Limit Example 4 USING THEOREM 2 Find the limit of (a) lim x (2 x ) , (b) 2 x 3 x lim x 2 Solution (a) lim x 2 (2 x ) 32 (2 3) 9 x 3 (b) lim x 2 2 x 2x 4 x 2 2 2 2(2) 4 22 3 2 2x 4 x 2 2.1 Rate of Change and Limits (12, Example 5) Properties of Limit Example 5 USING THE PRODUCT RULE tan x Find the limit of (a) lim x 0 x Solution lim x 0 tan x x lim x 0 lim x 0 sin x x sin x x 1 cos x 1 lim x 0 cos x 1 1 1 cos 0 2-1 Exercise 63 2 0 -4 -2 -2 0 2 4 2.1 Rate of Change and Limits (13, Example 6) Properties of Limit Example 6 EXPLORING A NONEXISTENT LIMIT Find the limit of x 3 1 lim x 2 x 2 Solution From graph, 200 it does not exist. 100 0 -100 -10 -5 0 5 10 2.1 Rate of Change and Limits (14) One-sided and Two-sided Limits Definition right-hand left-hand lim f ( x ) The limit of f as x approaches c x c from the right. lim f ( x ) The limit of f as x approaches c x c from the left. f(x) f(c-) f(c+) c x 2.1 Rate of Change and Limits (15, Example 7) One-sided and Two-sided Limits Example 7 FUNCTION VALUES APPROACH TWO NUMBERS Find the limit of f ( x ) int( x ) Solution lim int( x ) 0 5 lim int( x ) 1 3 x 1 x 1 4 2 1 0 -1 -2 -3 -2 -1 0 1 2 3 4 5 2.1 Rate of Change and Limits (16, Theorem 3) One-sided and Two-sided Limits Theorem 3 One - sided and Two - sided Limi ts A function f ( x ) has a limit as x approaches c if and only if the right-hand and left-hand limits at c exist and are equal. In symbols, lim f (x ) L lim f (x ) L x c x c and lim f ( x ) L x c 2.1 Rate of Change and Limits (17, Example 8) One-sided and Two-sided Limits Example 8 EXPLORING RIGHT- AND LEFT-HAND LIMITS Find the limit of x 1, 0 x 1 1, 1 x 2 lim f (x ) 0 x 1 x 2 x 1, 2 x 3 x 5, 3 x 4 f (x ) 3 2, lim f (x ) 1 x 1 lim f ( x ) 1 x 2 lim f ( x ) 1 x 2 2 lim f ( x ) 2 x 2 lim f ( x ) 2 1 x 3 x 4 x 3 x 4 lim f ( x ) 2 0 0 1 2 3 4 5 lim f (x ) 1 lim f ( x ) 2 x 3 lim f (x ) 1 2.1 Rate of Change and Limits (18, Theorem 4) Sandwich Theorem Theorem 4 The Sandwich Theore m If g (x ) f (x ) h (x ) for all x c in some interval about c , and lim g (x ) lim h (x ) L, then lim f ( x ) L x c x c x c y h L f g c x 2.1 Rate of Change and Limits (19, Example 9) Sandwich Theorem Example 9 USING THE SANDWICH THEOREM Show that lim x 2 sin(1/ x ) 0 x 0 Solution since and Finally, x 2 x 2 sin 1x x 2 lim( x 2 ) lim( x 2 ) 0 x 0 x 0 lim x 2 sin 1x 0 x 0 0.02 h(x) = x2 0.01 0 -0.01 f(x) = x2 sin(1/x) g(x) = -x2 -0.02 -0.2 -0.1 0 0.1 0.2 2.2 Limits Involving Infinite (1) Finite Limits as x→± • The symbol of infinite () does not represent a real number. • The use to describe the behavior of a function when the values in its domain or range out grow all finite bounds. The limit of f as x approaches infinite means the limit of f as x moves increasingly far to the right on the number line (x ). the limit of f as x moves increasingly far to the left on the number line (x ). The limit in each case may or may not exist. 2.2 Limits Involving Infinite (2) Finite Limits as x→± Looking at -6 f (x ) 1/ x -4 -2 0 2 4 6 (a) as x , (1/ x ) 0 and we write lim(1/ x ) 0 x (b) as x , (1/ x ) 0 and we write lim (1/ x ) 0 x 2.2 Limits Involving Infinite (3) Finite Limits as x→± Definition Horizontal Asymptote The line y b is a horizontal asymptote of the graph of a function y f (x ) if either lim f (x ) b x Looking at or lim f (x ) b . x f ( x ) 2 1/ x 1 1 lim 2 2 and lim 2 2 x x x x This function has the single horizontal asymptote y 2 2.2 Limits Involving Infinite (4, Example 1) Finite Limits as x→± Example 1 LOOKING FOR HORIZONTAL ASYMPTOTES x Investigate the graph of f ( x ) x 2 1 Solution From graph shows x lim 1 and 2 x x 1 x lim 1 2 x x 1 This function has the two horizontal asymptotes y 1 and y 1 1 0 -8 -4 0 -1 4 8 2.2 Limits Involving Infinite (5, Example 2) Sandwich Theorem Revisited Example 2 FINDING A LIMIT AS x APPROACHES sin x Find lim x x Solution since, -1 sinx 1, for x 0 -1 sin x 1 x x x by Sandwich theorem 1 sin x 1 -1 0 lim lim lim x x x x x x 0 -10 0 -1 10 2.2 Limits Involving Infinite (6, Theorem 5-1) Sandwich Theorem Revisited Theorem 5 Properties of Limits as x → ±∞ If L,M, and k are real numbers and lim f (x ) L and lim g (x ) M ,then x 1. Sum Rule : 2. Difference Rule : 3. Product Rule : 4. Constant Multiple Rule : x lim (f ( x ) g (x )) L M x lim (f ( x ) g ( x )) L M x lim (f ( x ) g ( x )) L M x lim (k f (x )) k L x 2.2 Limits Involving Infinite (7, Theorem 5-2) Sandwich Theorem Revisited Theorem 5 Properties of Limits as x → ±∞ If L, M , and k are real numbers and lim f ( x ) L and lim g ( x ) M , then x x f (x ) L 5. Quotient Rule : lim ,M 0 x g ( x ) M 6. Power Rule : If r and s are integers, s 0, then lim (f (x ))r s x provided that Lr s is a real number. Lr s 2.2 Limits Involving Infinite (8, Example 3) Sandwich Theorem Revisited Example 3 USING THEOREM 5 5x sin x Find lim x x Solution 5x sin x 5x sin x sin x since, 5 So, x x x x 5x sin x sin x lim lim 5 lim 5 x x x x x 6 4 2 -10 0 -2 -4 -6 0 10 2.2 Limits Involving Infinite (9, Exploration 1-1) Sandwich Theorem Revisited Exploration 1 Exploring Theorem 5 1. Let f (x ) 5x sin x , g (x ) x . Can we apply Quotient Rule to lim f ( x ) / g ( x )? Does the limit exist ? x 6 5 4 -60 -40 -20 0 20 40 60 2.2 Limits Involving Infinite (10, Exploration 1-2) Sandwich Theorem Revisited Exploration 1 Exploring Theorem 5 2. Let f (x ) sin2 x , g (x ) cos 2 x . Describe the behavior of f and g as x ? Can we apply the Sum Rule to lim(f (x ) g (x )) ? x 2 f(x) = sin2x g(x) = cos2x 1 0 0 5 10 15 20 25 2.2 Limits Involving Infinite (11, Exploration 1-3) Sandwich Theorem Revisited Exploration 1 Exploring Theorem 5 3. Let f (x ) ln(2x ), g ( x ) ln( x 1). Find the limit of f and g as x ? Can we apply the Difference Rule to lim(f (x ) g (x )) ? Does the limit exist x 4 3 2 1 f(x) = ln(2x) g(x) = ln(x+1) 0 -1 0 5 10 15 20 25 2.2 Limits Involving Infinite (12) Infinite Limits as x →a Looking at (a) as x 0 , (b) as x 0 , f ( x ) 1/ x lim (1/ x ) x 0 lim (1/ x ) x 0 -6 -4 -2 0 2 4 6 2.2 Limits Involving Infinite (13) Infinite Limits as x →a Definition Vertical Asymptote The line x a is a vertical asymptote of the graph of a function y f (x ) if either lim f (x ) or lim f (x ) . x a x a 2.2 Limits Involving Infinite (14, Example 4) Infinite Limits as x →a Example 4 FINDING VERTICAL ASYMPTOTES Find the vertical asymptotes f ( x ) Solution since, lim x 0 1 x 2 and lim x 0 1 x 2 1 x2 line x 0 is the only vertical asymptote. 4 3 2 1 0 -12 -8 -4 0 4 8 12 2.2 Limits Involving Infinite (15, Example 5) Infinite Limits as x →a Example 5 DINDING VERTICAL ASYMPT OTES Find the vertical asymptotes f ( x ) tan x Solution when a n , (n odd integer), 2 lim tan(x ) and lim tan x x a x a 4 2 0 -8 -4 -2 -4 0 4 8 2.2 Limits Involving Infinite (16, Example 6) End Behavior Models Example 6 MODELING FUNCTIONS FOR | x | LA RGE Let f (x ) 3x 4 2x 3 3x 2 5x 6 and g ( x ) 3x 4 Solution The graph f and g , quite different near the origin, are identical on a larger scale. f (x ) 3x 4 2x 3 3x 2 5x 6 lim 1 4 x g ( x ) 3x 20 40000 15 10 20000 5 0 0 -5 -2 -1 0 1 2 -20 -10 0 10 20 2.2 Limits Involving Infinite (17) End Behavior Models Definition End Behavior Model The function g is (a) a right end behavior model for f if and only if f (x ) lim 1 x g ( x ) (b) a left end behavior model for f if and only if f (x ) lim 1 x g ( x ) 2.2 Limits Involving Infinite (18, Example 7) End Behavior Models Example 7 FINDING END BEHAVIOR MODELS Let f ( x ) x e x , show that g ( x ) x is a right end behavior model for f while h ( x ) e x is a left end behavior models for f . Solution on the right e x f (x ) x e x lim lim lim 1 x g ( x ) x x x x 1 on the left f (x ) x e x x lim lim lim 1 x x x h ( x ) x x e e 8 1 4 0 -8 -4 0 4 8 2.2 Limits Involving Infinite (19) End Behavior Models • IF one function provides both a left and right behavior model, it called an end behavior model. • In general, g(x) = anxn ia an end behavior model for the polynominal function f(x) = anxn + an-1xn-1 +…+ ao . In the large, all polynominals behave like monomials. • This is the key to the end behavior of rational functions. 2.2 Limits Involving Infinite (20, Example 8) End Behavior Models Example 8 FINDING END BEHAVIOR MODELS Find the end behavior model for 2x 3 x 2 x 1 2x 5 x 4 x 2 1 , (b) f ( x ) (a) f (x ) 2 5x 3 x 2 x 5 3x 5x 7 Solution (a) end behavior of numerator is 2x 5 , demominator is 3x 2 , the end behavior model of f is 2 3 x3 (b) end behavior of numerator is 2x 3 , demominator is 5x 3 , the end behavior model of f is 2 5 2.2 Limits Involving Infinite (21, Example 9) End Behavior Models Example 9 FINDING A HORIZONTAL ASYMPTOTE 4x 2 3x 5 Dose the graph f ( x ) have a horizontal 3 2x x 1 symptote ? If so, what is it ? Solution The end behavior for f is 4x 2 / 2x 3 2 / x so, lim f (x ) lim 2 / x 0, x x y 0 is the horizontal asymptote of f . 8 4 0 -8 -4 0 4 8 2.2 Limits Involving Infinite (22, Example 10) Seeing Limits as x→± Example 10 USING SUBSTITUTION lim sin(1/ x ) Find x Solution replacing lim sin(1/ x ) by the equivalent lim sin(x ) 0 x then, lim sin(1/ x ) lim sin( x ) 0 x -8 -4 x 0 x 0 2 2 1 1 0 0 -1 -1 -2 -2 0 4 8 -0.08 -0.04 0 0.04 0.08 2.3 Continuity (1, Example 1-1) Continuity at a Point Example 1 INVESTIGATING CONTINUIT Y x 1, 1, f ( x ) 2, x 1, x 5, 3 0 x 1 1 x 2 x 2 2x 3 3x 4 Solution 2 This function f is continuous at every point in its domain 1 [0, 4], except at x 1 and x 2. 0 0 1 2 3 4 5 2.3 Continuity (2, Example 1-2) Continuity at a Point Example 1 INVESTIGATING CONTINUITY Solution point at which f is continuous x 0, lim f (x ) f (0) x 0 x 4, lim f (x ) f (4) x 4 x c , c 1, 2, lim f (x ) f (c ) 3 point at which f is discontinuous 2 x c x 1, x 2, lim f ( x ) does not exist x 1 lim f ( x ) 1, but 1 f (2) 1 x 2 0 0 1 2 3 4 5 2.3 Continuity (3) Continuity at a Point Definition Continuity at a Point Interior Point : A function y f (x ) is continuous at an interior point c of its domain if lim f (x ) f (c ) x c Endpoint : A function y f (x ) is continuous at a left endpoint a or is continuous at a right endpoint b of its domain if lim f (x ) f (a ) or lim f (x ) f (b ), respectively. x a x b Continuity from both side Continuity from the left Continuity from the right a c b 2.3 Continuity (4, Example 2) Continuity at a Point Example 2 FINDING POINT OF CONTINUITY & DISCONTINUITY Consider the integer function f ( x ) int( x ) Solution For the function to be continuous at x c , the limit as x c must exist 5 4 and must equal the value of 3 2 the function at x c . This function is discontinuous at every integer. lim (int x ) 2, x 3 lim (int x ) 3 x 3 1 0 -1 -2 -3 the limit as x n does not exist -2 -1 0 1 2 3 4 5 2.3 Continuity (5) Continuity at a Point 1 y = f(x) y = f(x) 1 continuous at x=0 1 2 y = f(x) continuous at x=0 If it had f(0)=1 continuous at x=0 If it had f(0)=1 continuity at x = 0 are removable 2.3 Continuity (6) Continuity at a Point lim f (x ) does not exist, no way to x 0 1 improve it. It is a trype of a jump discontinuity. y = f(x) 6 lim1/ x 2 , 4 x 0 an infinite discontinuity. 2 0 -4 -2 0 2 4 2.3 Continuity (7) Continuity at a Point 1 lim sin(1/ x ), x 0 0 -1 an oscillating discontinuity. 2.3 Continuity (8, Exploration1-1,2) Continuity at a Point Exploration 1 Removing a Discontinuity x 3 7x 6 Consider the function f (x ) x2 9 1. Factor the denominator. What is the domain of f (x ) 2. Investigate the graph of f around x 3 to see that f has a removable discontinuity at x 3 40 0 -4 -2 0 -40 2 4 2.3 Continuity (9, Exploration1-3) Continuity at a Point Exploration 1 Removing a Discontinuity x 3 7x 6 Consider the function f (x ) x2 9 3. How should f be defined at x 3 to remove the discontinuity ? 2.00 2.4000 2.90 3.2390 2.95 3.2861 2.99 3.3239 3.01 3.3428 3.05 3.3806 3.10 3.4279 4.00 4.2857 40 0 -4 -2 0 -40 2 4 2.3 Continuity (10, Exploration1-4) Continuity at a Point Exploration 1 Removing a Discontinuity Consider the function x 3 7x 6 f (x ) x2 9 4. Show that (x - 3) is a factor of the numerator of f , and remove all commond factors. Now computer the limit as x 3 of the reduced form f . 40 0 -4 -2 0 -40 2 4 2.3 Continuity (11, Exploration1-5) Continuity at a Point Exploration 1 Removing a Discontinuity Consider the function x 3 7x 6 f (x ) x2 9 5. Show that the extended function x 3 7x 6 , x 3 g (x ) x 2 9 10 / 3, x 3 is continuous at x 3. The function g ( x ) is the continuous extension of the original function f to include x 3. 2.3 Continuity (12) Continuous Functions • A function is continuous on an interval if and only if it is continuous at every point of the interval. • A continuous function is one that is continuous at every point of its domain. 2.3 Continuity (12, Example 3) Continuous Functions Example 3 IDENTIFYING CONTINUOUS FUNCTIONS Consider the function f ( x ) 1/ x Solution It is a continuous function because it is continuous at every point of its domain. However, it has a point of discontinuity at x 0 because it is not defined there. 4 2 0 -4 -2 -2 -4 0 2 4 2.3 Continuity (13) Continuous Functions • Polynominal functions f are continuous at every real number c because limx → c, f(x) = f(c). • Rational functions are continuous at every point of their domains. They have points of discontinuity at the zeros of their denominators. • The absolute value function y = |x| is continuous at every real number. • The exponential, logarithmic, trigonometric functions and radical function like y = x (1/n) (n = a positive integer greater than 1) are continuous at every point of their domains. 2.3 Continuity (14, Theorem 6) Algebraic Combinations Theorem 6 Properties of Continuous Function s If the functions f and g are continuous at x c , then the following combinations are continuous at x c . 1. Sums : f g 2. Difference : f g 3. Products : f g 4. Constant multiples : 5. Quotients : k f , for any number k f / g , provided g (c ) 0 2.3 Continuity (15, Theorem 7) Composites Theorem 7 Composite of Continuous Function s If f is continuous at c and g is continuous at f (c ), then the composite g f is continuous at c . g f continuous at c g (x ) sin x 2 continuous at f(c) c continuous at c f (x ) x 2 f(c) g(f(c)) g (f (x )) sin x 2 2.3 Continuity (16, Example 4) Composites Example 4 USING THEROREM 7 x sin x Show that y 2 is continuous x 2 Solution The function is continuous by letting x sin x g (x ) | x |, f (x ) 2 x 2 0.4 0.3 0.2 0.1 0 -8 -4 0 4 8 2.3 Continuity (17, Theorem 8) Intermediate Value Theorem for Continuous Functions Theorem 8 The Intermediate Value Theorem for Continuous Functions A functions y f ( x ) that is continuous on a closed interval a , b takes on every value between f (a ) and f (b ). In other words, if y 0 is between f (a ) and f (b ), then y 0 f (c ) for some c in a , b . f(b) yo f(a) a c b 2.3 Continuity (18, Example 5) Intermediate Value Theorem for Continuous Functions Example 5 USING THEROREM 8 Is any real number exactly 1 less than its cube ? 2.4 Rates of Change and Tangent Lines (1) Average Rates of Change • The average rate of change of a quantity over a period of time is the amount of change divided by the time it takes. Such as average speed, grows rate of populations, monthly rainfall. • In general, the average rate of change of a function over an interval is the amount of change divided by the length of the interval. y dy Rate of change dx f(x) dy dx x 2.4 Rates of Change and Tangent Lines (2) Average Rates of Change (Example 1) Example 1 FINDING AVERAGE RATE OF CHANGE Find the average rate change of f (x ) x 3 x over the interval [1, 3]. Solution rate of change f (3) - f (1) 3 -1 60 24 - 0 12 2 40 20 0 -20 -4 -2 0 2 4 2.4 Rates of Change and Tangent Lines (3) Average Rates of Change (Example 2-1) Example 2 GROWING DROSOPHILA IN A LABORATORY Ue the point P (23, 150) and Q (45, 340) to computer the average rate of change and the slope of the secant line PQ Solution average rate of change p 340 150 t 45 23 8.6 flies/day 2.4 Rates of Change and Tangent Lines (4) Average Rates of Change (Example 2-2) Example 2 GROWING DROSOPHILA IN A LABORATORY Ue the point P (23, 150) and Q (45, 340) to computer the average rate of change and the slope of the secant line PQ Solution the secant slope p 340 150 t 45 23 8.6 flies/day an average rate of change as the slope of a secant line 2.4 Rates of Change and Tangent Lines (5) Average Rates of Change (Example 2-3) Example 2 GROWING DROSOPHILA IN A LABORATORY In additional, we may want to know how fast the population was growing on day 23 itself. Solution to find out, we can watch the slope of the secant PQ change as we back Q along the curve toward P . Q (45,340) (40,330) slope PQ (340-150)/(45-23) 8.6 (330-150)/(40-23) 10.6 (35,310) (30,265) (310-150)/(35-23) 13.3 (265-150)/(30-23) 16.4 2.4 Rates of Change and Tangent Lines (6) Average Rates of Change (Example 2-4) Example 2 GROWING DROSOPHILA IN A LABORATORY In additional, we may want to know how fast the population was growing on day 23 itself. Solution In terms of geometry, as Q approaches P along the curve is : The secant PQ approaches the tangent line AB that we drew by eye at P . This means that within the limitations of our drawing, the slopes of the secants approach the slop of the tangent 350-0 17.5 flies/day 35-15 2.4 Rates of Change and Tangent Lines (7) Tangent to Curve • The rate at which the value of the function y = f(x) is changing with respect to x at any particular value x = a to be the slope of the tangent to the curve y = f(x) at x = a. • Are we to define the tangent line at an arbitrary point P on the curve and find its slope from the function y = f(x) ? Our usual definition of slope requires two points 2.4 Rates of Change and Tangent Lines (8) Tangent to Curve The procedure to find the slope of a point P is follows : 1. We start with what we can calculate, namely, the slope of a secant through P and a point Q nearby on the curve. 2. We find the limiting value of the secant slope (if it exists) as Q approaches P along the curve. 3. We define the slope of the curve at P to be this number and define the tangent to the curve at P to be the line through P with this slope. 2.4 Rates of Change and Tangent Lines (9) Tangent to Curve (Example 3-1) Example 3 FINDING SLOPE AND TANGENT LINE Find the slope of the parabola y x 2 at the point P (2,4). Write this tangent line equation. Solution consider point P (2, 4) and a nearby point Q (2 h , (2 h )2 ). Q(2+h, (2+h)2) P(2,4) -1 0 1 2 3 4 2.4 Rates of Change and Tangent Lines (10) Tangent to Curve (Example 3-2) Solution (y x 2 ) consider point P (2, 4) and a nearby point Q (2 h , (2 h )2 ). y secant slope x (2 h )2 4 h 4 4 secant slope is (2 h )2 4 h 4 4 Q(2+h, (2+h)2) tangent slope 4 y (2 h )2 4 P(2,4) x h -1 0 1 2 3 4 2.4 Rates of Change and Tangent Lines (11) Tangent to Curve (Example 3-3) Solution (y x 2 ) The limit of the secant slope as Q appooaches P along the curve is secant slope is lim (secant slope) lim(h 4) 4 Q P (2 h )2 4 h 4 4 h 0 the slope at P is 4 and the line equation y 4x - 8 Q(2+h, (2+h)2) tangent slope 4 y (2 h )2 4 P(2,4) x h -1 0 1 2 3 4 2.4 Rates of Change and Tangent Lines (12) Slope of a Curve Definition Slope of a Curve at a Point The slope of the curve y f ( x ) at the point P (a ,f (a )) is the number f (a h ) f (a ) , h 0 h m lim provided the limit exists. y y=f(x) Q(a+h, f(a+h)) f(a+h)-f(a) P(a, f(a)) h a a+h x 2.4 Rates of Change and Tangent Lines (13) Tangent to Curve (Example 4-a) Example 4 EXPLORING SLOPE AND TANGENT Let f (x ) 1/ x (a) Find the slope of the curve at x a (b) Where does the slope equal -1/4 ? (c) What happens to the tangent to the curve at the point (a , 1/ a ) for different values of a ? Solution (a) The slope at x a is 1 1 f (a h ) f (a ) 1 1 a h a lim lim lim 2 h 0 h 0 h 0 a (a h ) h h a 2.4 Rates of Change and Tangent Lines (14) Tangent to Curve (Example 4-b) Example 4 EXPLORING SLOPE AND TANGENT Let f ( x ) 1/ x (b) Where does the slope equal -1/4 ? Solution (b) The slope will be 1/ 4 if 1 1 2 a 4 a 2 -4 -2 0 2 4 2.4 Rates of Change and Tangent Lines (15) Tangent to Curve (Example 4-c) Example 4 EXPLORING SLOPE AND TANGENT Let f (x ) 1/ x (c) What happens to the tangent to the curve at the point (a , 1/ a ) for different values of a ? Solution The slope 1/ a 2 is always negtive. As a 0+ , the slope approaches - and the tangent becomes increasingly steep. As a move away from the origin in either direction, the slope approaches 0 and the tangent becomes inreasingly horizontal. -4 -2 0 2 4 2.4 Rates of Change and Tangent Lines (16) Tangent to Curve ALL of these are the same : the slope of y f ( x ) at x a the slope of the tangent to y f ( x ) at x a the (instantaneous) rate of change of f ( x ) with respect to x at x a f (a h ) f (a ) lim h 0 h 2.4 Rates of Change and Tangent Lines (17) Normal to a Curve (Example 5) Example 5 FINDING A NORMAL LINE Write an equation for the normal to the curve f (x ) 4 x 2 at x 1 Solution The slope at x 1 is f (1 h ) f (1) 4 (1 h )2 3 lim lim h 0 h 0 h h h (2 h ) lim 2 h 0 h The normal line at x 1 -3 -2 5 1 y 2x 2 -1 0 1 2 3 2.4 Rates of Change and Tangent Lines (18) Speed Revisited (Example 6) • A body’s average speed along a coordinate axis for a given period of time is the average rate of change of its position y = f(t). • Its instantaneous speed at ant time t is the instantaneous rate of change of position with respect to time at time t. Example 6 INVESTIGAING FREE FALL A falling rock position with f (t ) 16t 2 , Find the speed at t 1. Solution f (1 h ) f (1) 16(1 h )2 16(1)2 The speed is lim lim h 0 h 0 h h lim16(h 2) 32 h 0