2.1 Derivatives and Rates of Change

Transcription

1 1a 1b 2.1 Derivatives an Rates of Change Tangent Lines Example. Consier y f x x x-, 0 4 y-, f(x) axes, curve C Consier a smooth curve C. A line tangent to C at a point P both intersects C at P an has the same slope as C at P. a line t The Tangent Line Problem Given point P on curve C, how o you fin the tangent line? What is the equation of the line tangent to the curve at x 1? a P, line Point-slope form for a straight line passing through P(1,1) y 1 m(x 1)

2 2a 2b What is the slope m? What is the slope of the secant passing through P(1,1) an Q 1 2,4? a Q 1, PQ 1 m PQ1 Δy Δx f 2 f(1) What is the slope of the secant passing through P(1,1) an Q 2 1.5, f 1.5? a Q 2, PQ 2 m PQ2 f 1.5 f(1) m PQ f(x) f(1) x 1 this ratio is calle a ifference quotient x 2 1 x 1 (x 1)(x+1) x 1 x + 1, x 1 unefine, x 1 As long as x 1, m PQ x + 1 x m PQ The slope of the tangent line is the limit of the ifference quotient as x 1. m lim x 1 x What is the slope of the secant line passing through P(1,1) an Q(x, x 2 )? The equation of the tangent line is y 1 2(x 1)

3 3a 3b Example. Fin an equation of the line tangent to the curve g x x + 1 at P(3,2). Point slope form of the tangent line y 2 m(x 3) where m lim x 3 g x g(3) x 3 simplify the ifference quotient g x g(3) x 3 x+1 2 x 3 multiply by 1 to rationalize the numerator m lim x 3 1 x Equation of tangent line y 2 1 (x 3) 4 The Velocity Problem Drive to Spokane airport (~85 miles) Start at noon Drive slowly through Colfax Have lunch at Harvester x+1 2 x 3 x+1 4 x+1+2 x+1+2 x 3 ( x+1+2) cancel factors of x 3 Thus 1 x+1+2 true if x 3 Arrive 2pm Average velocity 42.5 mph istance travele time elapse 85 miles 2 hours The speeometer 5 miles north of Colfax reas 65 mph. This is the instantaneous velocity. Mathematical efinition of instantaneous velocity?

4 4a Galileo rops a ball off the leaning Tower of Pisa sketch groun, tower, coorinate s with origin at top What is the average velocity between t 1 an t 2? 4b Average velocity s 2 s(1) meters secon istance travele time elapse ball falls istance s at time t after release. s(t) 5t 2 s meters, t secons 0 2 t-, s-, curve What is the average velocity between t 1 an a variable t? Average velocity s t s(1) t 1 5 t 2 5 t 1 5 (t 2 1) t 1 5 t 1 (t+1) t 1 5(t + 1) as long as t 1.

5 5a 5b Table t average velocity (meters per sec.) Derivatives Define the erivative of a function f at a number a, enote f (a) f a lim x a f x f(a) x a [1] Define instantaneous velocity at t 1 as the limit of average velocities over shorter an shorter time intervals aroun t 1. Denote instantaneous velocity v(t). From the example above v 1 s 1 lim t 1 s t s(1) t 1 Alternatively, introuce. v 1 lim t 1 s t s(1) t 1 x a x a + lim t 1 5(t + 1) 10 meters secon an insert in equation [1] to get f a lim 0 f a+ f(a) [2]

6 6a 6b 2.2 The Derivative as a Function Graph an compare f(x) an f (x) x-, y-, f(x) Replace the symbol a in [2] by x. Regar x as a variable. f x lim 0 f x+ f(x) Regar f as a new function. Example. Let f x x 2. Fin f (x) x-, y-, f (x) Simplify the ifference quotient f x+ f(x) x+ 2 x 2 x 2 +2x+ 2 x 2 2x+ 2 2x + this step assumes 0 Then f x lim 0 2x + 2x a tangent segments at x 1, 0, 1 to graph of f a ots at x 1, 0, 1 to graph of f

7 7a 7b Example. Let g x x + 1. Fin g (x).?? Transparency: match f an f Simplify the ifference quotient g x+ g(x) x++1 x+1 rationalize numerator by multiplying by 1 x++1 x+1 multiply terms in numerator x++1 (x+1) x++1+ x+1 x++1+ x+1 x++1+ x+1 ivie through by (assumes 0) Then 1 x++1+ x+1 g x lim 0 1 x++1+ x+1 Notations for Derivative original function: y f(x) erivative function: f x y prime notation emphasizes iea of erivative as a new function f (x) the prime means ifferentiate with respect to function argument evaluate at no. a f (a) 1 2 x+1 Show transparency comparing g an g

8 8a 8b Leibniz notation emphasizes iea of erivative as the limit of a ratio x x +,, f, Δx, f x + f x Δy Operator notation for erivative Sometimes we write f x f or f x Df view an D as operators: machines that convert the functions they operate on into other functions lim 0 f x+ f(x) Δy lim 0 y Δx evaluate at no. a y a Differentiability f ifferentiable at a means f (a) exists f ifferentiable on an open interval (a, b) means f is ifferentiable at every point in (a, b)

9 9a 9b Example (a function not ifferentiable at a point) f x x Is f ifferentiable at x 0? Geometrical Iea axes, graph of x, no tangent line here (at origin) if so f 0 lim 0 f 0+ f(0) f lim h lim 0 oes this limit exist? fin the limit from the right lim 0+ lim 0 1 fin the limit from the left lim 0 lim 0 1 the right an left han limit o not agree. conclue f (0) oes not exist To be ifferentiable at point, the graph must have a unique tangent line at that point.

10 10a 10b Three ways that a function can fail to be ifferentiable (c) at a vertical tangent a,, function w/ a vertical tangent at a (a) at any iscontinuity a,, function with cty at a lim x a f x, f (a) DNE lim 0 f a+ f(a) DNE (b) at any corner or kink a,, function with a kink at a lim 0 f a+ f(a) DNE

11 11a Relationship between ifferentiability an continuity We have shown: if f is not continuous then f is not ifferentiable Let A be the statement f is continuous at a no. x Let B be the statement f is ifferentiable at x We have shown If (not A) then (not B) This is logically equivalent to If B then A Higher Derivatives Consier y f(x) First erivative of f y f (x) 11b Regare as a function, f may itself be ifferentiable. Secon erivative of f 2 y 2 f (x) If f (x) is ifferentiable, form the thir erivative 3 y 3 f (x) If f (x) is ifferentiable, form the fourth erivative 4 y 4 f 4 (x) Notation for the n T erivative, with n 4: If f is ifferentiable at x then it is continuous at x n y n f n (x)

12 12a 12b Application of higher erivatives 2.3 Basic Differentiation Rules Let s(t) be the position of an object at time t. s t s t v(t) is the velocity of the object We first consier those rules that will enable us to ifferentiate polynomials. Derivative of a Constant Function x-, 0 c y-, line y c s t v t a(t) is the acceleration of the object First an secon erivatives are the most important in applications Example. Mechanics slope of tangent line? c 0 Derivative of f x x 0 x-, 0 y-, y x momentum mass velocity force mass acceleration slope of tangent line? x 1

13 13a 13b Derivative of g x x 2 We have seen that g x 2x The Power Rule. Let n be a positive integer xn nx n 1 Derivative of F x x 3 Example. x7 7x 6. F x lim 0 F x+ F(x) simplify the ifference quotient F x+ F(x) thus x+ 3 x 3 x 3 +3x 2 +3x x 3 3x 2 +3x x 2 + 3x + 3 assumes 0 F x lim 0 3x 2 + 3x + 2 3x 2 Proof. Preliminary fact: x a (x n 1 + x n 2 a + x n 3 a xa n 2 + a n 1 ) x n + x n 1 a + x n 2 a xa n 1 x n 1 a x n 2 a 2 xa n 1 a n x n in other wors x n a n x a xn 1 + x n 2 a + xa n 2 + a n 1 a n notice there are n terms on the right han sie

14 14a 14b Let f x x n f x a lim n a n x a x a lim x a (x n 1 + x n 2 a + xa n 2 + a n 1 ) a n 1 + a n 2 a + aa n 2 + a n 1 na n 1 Regar a as a variable. Replace a by x. f x nx n 1???? x 1 2 x 1 2 recall x x 1 2 t 1 t u u 2 The Constant Multiple Rule Let c be a constant an f a ifferentiable function cf x c f(x) The Power Rule (general version) Let n be any real number. xn nx n 1 a constant passes through the limit symbol Examples. 5x2 5 x2 5 2x 10x?? t 10 7 t Examples. x 9 9x 10 t t 2 2t 3?? w 5 w π

15 15a 15b The Sum Rule If f an g are both ifferentiable The Difference Rule If f an g are both ifferentiable f x + g x f x + g(x) f x g x f x g(x) In wors: the erivative of a sum is the sum of the erivatives prime notation f x + g x f x + g (x) shorthan f + g f + g the sum rule applies to the sum of any number of functions f + g + f + g + In wors: the erivative of a ifference is the ifference of the erivatives prime notation f x g x f x g (x) shorthan f g f g Example.?? x3 5x 2 Example x3 + 5x 2 + π 3x x + 0

16 16a We can now ifferentiate any polynomial Example. Let g x 5x 8 2x then g x 5x8 2x x8 2 x x 7 2 5x x 7 10x 4 We can ifferentiate other functions too. Example. Let 3 u t 2 Fin u t + 2 t 3 16b Example. Fin an equation of the line tangent to the curve y 3x x at the point (1,7) point slope form for straight line y y 0 m(x x 0 ) where x 0, y 0 (1,7) what is m? y 3 2x + 4 x 2 6x 4x 2 m y x answer y 7 2(x 1) u t t 3 2 u 2 3 t 3 t t t t 2

17 17a 17b?? Example. A ball is thrown straight up from the groun at 20 meters/secon. Its height is given by y 20 t 5 t 2 (a) Fin the velocity at time t. (b) Fin the velocity at t 1sec. (c) When is the ball at rest? () What is the average velocity between t 1 an t 2? Economics Marginal cost C x cost to prouce x wigets average rate of change of cost C x 2 C(x 1 ) x 2 x 1 marginal cost lim Δx 0 ΔC Δx C (x) ΔC Δx Example. Jeans manufacture. Let C x cost of proucing x pairs of jeans. where x x 2 $2000 capital costs (sewing machines) 3x x 2 cost of labor, materials, rent Cost of proucing 100 pairs of jeans C What is the cost of proucing one aitional pair of jeans? C x x x C x x x 2 C x + 1 C x x x (2x + 1)

18 18a 18b Cost of proucing the 101 ST pair C 101 C (201) Recall the aition formula for cosine cos a + b cos a cos b sin a sin (b) $5.01 Now use the limit efinition of erivative Compare with the marginal cost at 100 th pair C x x C 100 $5 cos x lim 0 cos x+ cos (x) aition formula for cosine lim 0 cos x cos sin x sin cos (x) C (x) is often a very goo approximation to the cost of proucing one aitional wiget. lim 0 cos x cos 1 ifference law for limits sin () lim 0 sin (x) constant multiple law of limits Derivatives of Sine an Cosine Recall the limits lim x 0 sin (x) x lim x 0 cos x 1 x 1 0 cos x lim 0 cos 1 recalling the limits above sin x lim 0 sin () cos x 0 sin x 1 sin (x)

19 19a 19b The erivative of sin (x) may be foun using a similar argument (see our text). In summary: sin x cos x cos x sin (x)?? Differentiate the following 1. f t sin t + π cos (t) 2.4 The Prouct an Quotient Rules Prouct Rule If f an g are both ifferentiable f x g x f x g x + g x f(x) or alternately (as I personally prefer) f x g x f x g x + f x g x prime notation f x g x f x g x + f x g (x) shorthan fg f g + fg 2. g y A + B cos(y) y 10 WARNING: fg f g The erivative of a prouct is not the prouct of erivatives This is a common mistake!

20 20a Example. By the power rule x3 3x 2 Now let f x 2 an g x then f 2x an g 1 x3 (fg) f g + fg 2x x + x 2 1 3x 2 Proof of the Prouct Rule Suppose f an g are both ifferentiable functions. Let f x g x F(x) then f x g x F(x) lim 0 F x+ F(x) lim 0 f x+ g x+ f x g(x) 20b subtract an a the same term in the numerator lim 0 f x+ g x+ f x+ g x +f x+ g x f x g(x) g x + g(x) lim[ f x + 0 f x + f(x) + g x ] algebra sum an prouct laws for limits lim 0 f x + lim 0 g x + g(x) + lim 0 g x lim 0 f x+ f(x) continuity of f an efinition of erivative f x g x + g x f (x)

21 21a 21b Prouct rule Extension to a Prouct of Three Functions If f, g an are all ifferentiable F x x 1 sin x + x 1 sin x x 2 sin x + x 1 cos(x) f g f g + f g + f g sin (x) x 2 + cos (x) x Example. Let f x g x x x then f g 1 x3 f g + f g + f g 1 x x + x 1 x + x x 1 3x 2 Example. Differentiate F x Law of exponents: 1 x x 1 sin (x) x. Quotient Rule If f an g are ifferentiable at a point x where g x 0 then f(x) g(x) shorthan f g f g fg g 2 f x g x f x g(x) g x 2 terms in numerator in same orer as my prouct rule (but take ifference) Then F x x 1 sin (x) Proof of quotient rule.

22 22a Let f g Then f g By prouct rule f g + g Solve for g f g f g g f g g f g 2 Example. Differentiate G x x+1 x 1. G x x+1 x 1 x+1 x 1 (x 1) 2 where x + 1 is shorthan for (x + 1) x 1 (x+1) (x 1) 2 2 x b Example. Fin the equations of the tangent lines to the curve G x (x+1) (x 1) that are parallel to the line x + 2y 2. Solution. Parallel means same slope. Slope of line? 2y 2 x y x slope is m 1 2 where oes G have slope 1 2? G 2 1 x x x x 1 or x 3 solve for x form of equation for tangent line

23 23a 23b y y 0 m(x x 0 ) Consier x 0 1. y 0 G y 1 2 x + 1 Consier x 0 3. y 0 G y 2 1 (x 3) 2?? Class practice prouct an quotient rules