Calculus : 2-credit version
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1 1 / 50 Calculus : 2-credit version Tan s textbook : 7th edition Chapter 3 Hua-Huai, Félix, Chern Department of Computer Science National Taiwan Ocean University September 15, 2009
2 Calculus of several variables We shall learn the contents in this chapter : Rules and higher Order Derivatives Trigonometric functions Implicit differentiation Related Rates & Differentials Marginal Functions 2 / 50
3 Basic Differentiation Rules 3 / d c D 0 (c: a constant) dx f.x/ D 5 H) d dx f.x/ D f 0.x/ D 0: 2. d dx xn D nx n 1 (n 2 R) f.x/ D x 7 H) f 0.x/ D 7x 6 : Case. n D 3 Ch3-exs.jnt
4 Basic Differentiation Rules 4 / d d.cf.x// D c f.x/ (c: a constant) dx dx f.x/ D 3x 8 H) f 0.x/ D 3 8x 7 D 24x 7 : 4. d dx Œf.x/ g.x/ D f 0.x/ g 0.x/ f.x/ D 7 C x 12 H) f 0.x/ D 0 C 12x 11 D 12x 11 :
5 Basic Differentiation Rules 5 / Production rule. d dx Œf.x/ g.x/ D f 0.x/ g.x/ C f.x/ g 0.x/ f.x/ D x 3 C 2x C 5 3x 7 8x 2 C 1 H) f 0.x/ D 3x 2 C 2 3x 7 8x 2 C 1 Derivative of first C x 3 C 2x C 5 21x 6 D 30x 9 C 48x 7 C 105x 6 40x 4 45x 2 80x C 2: 16x Derivative of second
6 Example Find the slope and an equation of the tangent line to the graph of f.x/ D 2x C 1= p x at the point.1; 3/. (# 6 p. 165) Solution. Ch3-exs.jnt Find the derivative of the function f.x/ D x 3 p x C 1. (# 2 p. 174). Solution. Ch3-exs.jnt 6 / 50
7 Basic Differentiation Rules 7 / Quotient rule. d f.x/ D f 0.x/ g.x/ f.x/ g 0.x/ dx g.x/ g.x/ 2 Sometimes remembered as: d hi lo dœhi hi dœlo D : dx lo lo 2 f.x/ D 3x C 5 x 2 2 H) f 0.x/ D 3.x2 2/ 2x.3x C 5/ D 3x2 10x 6.x 2 2/ 2.x 2 2/ 2
8 Example 8 / 50 Find the derivative of the function h.x/ where (# 5, p. 176). Solution. h 0.x/ D D h.x/ D p x 1 C x 2.1 C x 2 / d p p d x x dx dx 1 C x2 1 C x 2 2 Ch3-exs.jnt
9 Other rules 9 / Chain rule If h.x/ D g.f.x// D.g ı f /.x/, then h 0.x/ D g 0.f.x// f 0.x/. Note. h.x/ D g.f.x// is a composite function. Another version Leibniz Notation. Let u D f.x/ and y D h.x/, then y D h.x/ D g.u/ and dy dx D dy du du dx.
10 Power rule 10 / Chain rule If h.x/ D f n.x/ D.f.x// n ; n 2 R, then h 0.x/ D n.f.x// n 1 f 0.x/. Find the derivative of f.x/ D p 3x 2 C 4x. Solution. Write f.x/ D 3x 2 C 4x 1=2. f 0.x/ D D 1 2 3x2 C 4x 1=2 d dx 3x2 C 4x 3x C 2 p 3x2 C 4x :
11 Example 11 / 50 Find the derivative of G.x/ D Solution. 2x 1 6 G 0.x/ D 7 3x C 5 2x 1 6 D 7 3x C 5 d dx 2x x C 5 2x 1 3x C x 1/6 D.3x C 5/ 2.3x C 5/ : 8
12 Example 12 / 50 Find dy dx where y D u5=2 and u D 7x 8 C 3x 2. Solution. dy dx D dy du du dx D 5 2 u3=2 56x 7 C 6x D 5 2 7x8 C 3x 2 3=2 56x 7 C 6x D 7x 8 C 3x 2 3=2 140x 7 C 15x :
13 More Examples 13 / 50 Differentiate the function G.x/ D p 1 C x 2. (# 2 p.188) Solution. Ch3-exs.jnt Find the slope of the tangent line to the graph of the function 2x C 1 3 f.x/ D. (# 6 p.189) 3x C 2 Solution. Ch3-exs.jnt Find d dx B.A.x// p A.x/B.x/!ˇˇˇˇxD1 when A.1/ D 1, A 0.1/ D, B.1/ D p 2 and B 0.1/ D p 3. Solution. Ch3-exs.jnt
14 Higher Derivatives 14 / 50 The second derivative of a function f is the derivative of the derivative of f at a point x in the domain of the first derivative. Derivative Second Third Fourth nth Notations f 00.x/ f 000.x/ f.4/.x/ f.n/.x/ d 2 y dx 2 d 3 y dx 3 d 4 y dx 4 d n y dx n
15 Example 15 / 50 Given f.x/ D 3x 5 2x 3 C 14, find f 000.x/. Solution. H) f 0.x/ D 15x 4 6x 2 H) f 00.x/ D 60x 3 12x H) f 000.x/ D 180x 2 12 Given f.x/ D.2x C 1/=.3x 2/, find f 00.2/. Solution. H) f 0 2.3x 2/ 3.2x C 1/.x/ D.3x 2/ 2 D 7.3x 2/ 2 H) f 00.x/ D 14.3x 2/ D.3x 2/ 3 H) f 000.2/ D f 00.x/ ˇ D 42 ˇxD2 4 3 D 21 32
16 Trigonometric functions We shall learn the contents in this section : Measurement of Angles The Trigonometric Functions Differentiation of Trigonometric Functions 16 / 50
17 Measurement 17 / 50 y Angle Radian Degree Œ0; 2 Œ0; 360 ı x Degree x ı Radian
18 18 / 50 Trigonometric function y x D cos : cosine function y D sin : sine function x P.x; y/ y x Other functions : tan D sin cos sec D 1 cos cot D cos sin csc D 1 sin
19 Graphs & Identities Identities Pythagorean identities sin 2 x C cos 2 x D 1 1 C cot 2 x D csc 2 x 1 C tan 2 x D sec 2 x Addition/Difference identities Double-angle identities sin 2x D 2 sin x cos x cos 2x D cos 2 x sin 2 x D 2 cos 2 x 1 D 1 sin.x y/ D sin x cos y cos x sin y cos.x y/ D cos x cos y sin x sin y 2 sin 2 x 19 / 50
20 Example Verify the identity : 2 sin.csc sin / D 1 C cos 2: Solution. Ch3-exs.jnt Squeeze Theorem. If f.x/ g.x/ h.x/ for all x near c, except possibly at c, then, if lim f.x/ D lim h.x/ D L H) lim g.x/ D L: x!c x!c x!c 20 / 50
21 Limit and continuity 21 / 50 y t sin t tan t The trigonometric functions are continuous. x As t approaches to 0, lim sin t D 0 : t!0 cos t D 1 : lim t!0 Conclusion. lim sin t D sin : t! cos t D cos : lim t! Proof. Ch3-exs.jnt
22 Proof 22 / 50 y For any 0 < t <, we have 2 the fact that t < 0 < sin t < t < tan t: By Squeeze theorem, t sin t tan t x lim t D lim. t/ D 0; t!0 t!0 sin t D 0: lim t!0
23 Differentiation 23 / 50 Lemma 1. lim t!0 sin t t Proof. As 0 < t < 2, 1 cos t D 1: 2. lim t!0 t D 0: sin t < t < tan t H) cos t < sin t < 1 t sin t H) lim D 1: t!0 C t by Squeeze Theorem. Along with sin t being even, so we t prove Ch3-exs.jnt
24 Rules 24 / 50 d d 1. sin x D cos x, dx dx sin.f.x// D cos.f.x// f 0.x/; d d 2. cos x D sin x, cos.f.x// D dx dx sin.f.x// f 0.x/; d 3. dx tan x D d sec2 x, dx tan.f.x// D sec2.f.x// f 0.x/; d 4. sec x D sec x tan x, dx d dx sec.f.x// D sec.f.x// tan.f.x// f 0.x/; Proof of 1 & 3. Ch3-exs.jnt
25 Example 25 / 50 Find the equation of tangent line to the graph at the point.=8; 1/. Solution. y D f.x/ D tan 2x Compute f 0.x/; Compute the slope : m D f 0 ; 8 Use the slope-point form to find the equation. Ch3-exs.jnt
26 Implicit differentiation 26 / 50 For the following two equations : x 2 y C y x 2 C 1 D 0 y D x2 1 x 2 C 1 y 3 C 7y D x 3 : Can we find the tangent line at each point on the graphs defined by these two equation? Look at their graphs first. Ch3.mws
27 Implicit differentiation : the difference 27 / 50 y D 3x 3 4x C 17 y is explicitly a function of x. y 3 C xy D 3x C 1 y is implicitly a function of x. To differentiate the implicit case we write f.x/ in place of y to get : Œf.x/ 3 C x Œf.x/ D 3x C 1:
28 Implicit differentiation : cont 28 / 50 Now differentiate Œf.x/ 3 C x Œf.x/ D 3x C 1 w.r.t x using the chain rule : Which means 3f.x/ 2 f 0.x/ C f.x/ C xf 0.x/ D 3: 3y 2 y 0 C y C xy 0 D 3 subbing in y.3y 2 C x/y 0 D 3 y Solve for y 0 H) y 0 D 3 y 3y 2 C x :
29 Examples. 29 / 50 Find dy dx for the equation (# 2, p. 222.) Solution. Ch3-exs.jnt y 3 y C 2x 3 x D 8: Find dy ˇ for the equation dx ˇ.x;y/D.1;2/ (# 4, p. 224.) Solution. Ch3-exs.jnt x 2 y 3 C 6x 2 D y C 12:
30 Implicit function theory 30 / 50 y f 0.x/.x; f.x// View the piece as a implicit functiondifferentiation y D f.x/ implicit applicable differentiation applicable../ / x
31 Related Rates Look at how the rate of change of one quantity is related to the rate of change of another quantity. Example. Two cars leave from an intersection at the same time. One car travels north at 35 mi./hr., the other travels east at 60 mi./hr. How fast is the distance between them changing after 2 hours? Note. The rate of change of the distance between them is related to the rate at which the cars are traveling. 31 / 50
32 Steps Steps to solve a related rate problem : 1. Assign a variable to each quantity. Draw diagram if appropriate. 2. Write down known values/rates. 3. Relate variables with an equation. 4. Differentiate equation implicitly. 5. Plug in values and solve. 32 / 50
33 Example 33 / 50 Two cars leave from an intersection at the same time. One car travels north at 35 mi./hr., the other travels east at 60 mi./hr. How fast is the distance between them changing after 2 hours? y x DistanceD z dx dy D 60, D 60 dt dt x D 120, y D 70 z D 10 p 193 x 2 C y 2 D z 2 H) 2x dx dt C 2y dy dt H) C D 2 10 p 193 H) dz dt D 965 p :5 D 2z dz dt dz dt
34 Example The supply equation for the price and quantity for audio tapes is x 2 3xp C p 2 D 5: Find the change rate of supply of tapes when the price p D 11 units and the quantity x D 4 (thousands). (x: quantity, p: price) Solution.Ch3-exs.jnt 34 / 50
35 Differentials : approximation 35 / 50 y y D f.x/ P x x C x x
36 Increments and differentials 36 / 50 An increment in x represents a change from x 1 to x 2 and is defined by : x D x 2 x 1. ( read as delta x ) An increment in y w.r.t x and x represents a change in y and is defined by : y D f.x C x/ f.x/. Let y D f.x/ be a differentiable function, then the differential of x, denoted dx, is such that dx D x. The differential of y, denoted dy, is dy D f 0.x/x D f 0.x/ dx. Note. y measures the actual change in y dy measures the approximate change in y
37 Differentials by picture 37 / 50 y y D f.x/ y P dy f.x C x/ f.x/ x x C x x
38 Example 38 / 50 Given f.x/ D 3x 2 x. Find 1. x as x changes from 3 to 3:02. x D 3:02 3 D 0: y and dy as x changes from 3 to 3:02. y D f.3:02/ f.3/ D 24: D 0:3412 dy D f 0.x/ dxˇ D.6x ˇxD3; dxd0:02 D 17 0:02 D 0:34: 1/ dxˇ ˇxD3; dxd0:02
39 Examples Approximate the value of p 26:5 by using differential. (# 4, p. 235) Solution. Ch3-exs.jnt Write p 26:5 D p 25 C 1:5 H) x D?; dx D?. Approximate the values of sin.42 ı / and cos.31 ı / by using differential. Solution. Ch3-exs.jnt Write 42 ı D? H) x D?; dx D?. 39 / 50
40 Examples 40 / 50 Estimating errors in measurement. The side of a cube measured with a maximum percentage error 2%. Estimate the one for its volume. Solution. Ch3-exs.jnt
41 Marginal Functions 41 / 50 Marginal Cost is the actual cost incurred of producing an additional unit. Denote it by C.x/. Marginal Cost Function approximates the change in actual cost of producing an additional unit. C.x C 1/ C.x/ C 0.x/ 1 Average Cost Function is the average cost function with respect to the number of units produced. C.x/ D C.x/ x
42 Marginal Functions 42 / 50 Marginal Average Cost Function measures the rate of change of the average cost function with respect to the number of units produced. C 0.x/ Marginal Revenue Function measures the rate of change of the revenue function; approximates the revenue from the sale of an additional unit. Marginal Profit Function measures the rate of change of the profit function; approximates the profit from the sale of an additional unit.
43 Marginal Revenue and Profit Functions For a revenue function, R.x/, the marginal revenue function is R 0.x/. For a profit function, P.x/, the marginal profit function is P 0.x/. Example. The monthly demand for T-shirts is given by p D 0:05x C 25.0 x 400/ where p denotes the wholesale unit price in dollars and x denotes the quantity demanded. The monthly cost function for these T-shirts is C.x/ D 0:001x 2 C 2x C 200: 43 / 50
44 Questions and solution Questions. 1. Find the revenue and profit functions. 2. Find the marginal cost, marginal revenue, and marginal profit functions. 3. Find the marginal average cost function. Solution. 1. Find the revenue and profit functions. Revenue D xp D x. 0:05x C 25/ D 0:05x 2 C 25x: Profit D revenue cost D 0:05x 2 C 25x. 0:001x 2 C 2x C 200/: D 0:049x 2 C 23x / 50
45 Questions and solution 45 / 50 Solution. (cont.) 2. Find the marginal cost, marginal revenue, and marginal profit functions. Marginal Cost D C 0.x/ D 0:002x C 2 Marginal Revenue D R 0.x/ D 0:1x C 25: Marginal Profit D P 0.x/ D 0:098x C 23: 3. Find the marginal average cost function. Average cost D C.x/ D 0:001x C 2 C 200 x Marginal average cost D C x/ D 0:001 x : 2
46 Example # 5,6 (p ) Model F speaker: Cost : C.x/ D 100x C 200; 000 Demand function p D 0:02x C 400, 0 x 20; 000. a) Find the revenue function R.x/, the profit function P.x/. b) Find the marginal revenue and profit functions. Solution. Ch3-exs.jnt 46 / 50
47 Elasticity of Demand 47 / 50 Demand function p D f.x/ : the quantity demand decreases as the price increases. ( x : quantity, p : the price) Viewing x as a function of p : x D F.p/ Percentage of change in the quantity demand Percentage of change in the unit price 1 D D F.p/.F.p C h/ F.p//= 1 p F.p/ =p.f.p C h/ F.p//.p C h p/ h
48 Elasticity of Demand (cont) If F is differentiable at p, then F.p C h/ h F.p/ F 0.p/: Conclusion. If f is a differentiable demand function defined by its inverse x D F.p/, then the elasticity of demand at price p is given by E.p/ D pf 0.p/ F.p/ : Demand is Elastic if E.p/ > 1; Unitary if E.p/ D 1; Inelastic if E.p/ < / 50
49 Elasticity of Demand : III 49 / 50 If the demand is elastic at p, then an increase in unit price causes a decrease in revenue. A decrease in unit price causes an increase in revenue. If the demand is unitary at p, then with an increase in unit price the revenue will stay about the same. If the demand is inelastic at p, then an increase in unit price causes an increase in revenue. A decrease in unit price causes a decrease in revenue. Explanation : R.p/ D xp D pf.p/ R 0.p/ D F.p/ C pf 0.p/ D F.p/ 1 C pf 0.p/ F.p/ D F.p/.1 E.p// :
50 Example # 7,8 (p. 209, 211) Demand equation p D 0:02x C 400, 0 x 20; 000, for some model of loudspeaker a) Find E.p/. b) Is demand elastic, unitary, or inelastic when p D 100? p D 300? c) If p D 100, will raising the unit price slightly cause the revenue to increase or decrease? Solution. Ch3-exs.jnt 50 / 50
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