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Chapter 3 14 - 164 39 40 41 42 CHAPTER 3 DlFFERENTlATION...

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Unformatted text preview: 164 39. 40. 41. 42. CHAPTER 3 DlFFERENTlATION RULES If P(t) denotes the population at time t and A(t) the average annual income, then T(t) : P(t)A(t) is the total personal income. The rate at which T(t) is rising is given by T’(t) = P(t)A/(t) + A(t)P’(t) => T'(1999) = P(1999)A/(1999) + A(1999)P’(1999) : (96l.400)($l400/yr) + ($30.593)(9200/y1‘) = $1,345,960.000/yr + $281.455.600/yr = $1,627,415.600/yr So the total personal income was rising by about $1.627 billion per year in 1999. The term P(t)A’(t) % $1.346 billion represents the portion of the rate of change of total income due to the existing population‘s increasing income The term A(t)P’(t) z $281 million represents the portion of the rate of change of total income due to increasing population. (a) f (20) : 10.000 means that when the price of the fabric is $20/yard, 10,000 yards will be sold. f '(20) : —350 means that as the price of the fabric increases past $20/yard, the amount of fabric which will be sold is decreasing at a rate of 350 yards per (dollar per yard). (b) ROD) = NO?) => 3'0?) : pf'(p) +f(P) - 1 => R’(20) : 20f’(20) + f(20) - 1 : 20(7350) + 10000 = 3000. This means that as the price of the fabric increases past $20 / yard. the total revenue is increasing at $3000 / ($ / yard). Note that the Product Rule indicates that we will lose $7000 / (33 / yard) due to selling less fabric. but that that loss is more than made up for by the additional revenue due to the increase in price. . 17 (ac + 1)(1) ! $(1) 1 ! . If y = f(a:) : a: + 1, then f’(;1;) # (z + 1)2 : (a: + 1)? When :3 _ a, the equation of the tangent 1 1 line is y — a: 1 : W01: — a). This line passes through (1,2) when 2 — a f: 1 : WU - a) 41> 2(a+1)2—a(a+1)=1—a 4:) 2a2+4a+2—a2—a—1+a:0 <:> a2+4a+1=0. #4:: 42 —4(1)(1): *4 33m — —2ix/§. 2(1) 2 6 —2i\/§ _ ~2i\/§.#1$\/§ f(*2-:“§)=’—”_2tf--r:m -1242; W‘ ~6‘fl6 2:2f:\/§—3A—1i\/§_1:L\/§ ' 1—3 _ —2 2 ‘ 6 the lines touch the curve at A(—2 + fl. 1%3> % (#027. —0.37) and V The quadratic formula gives the roots of this equation as a ! so there are two such tangent lines. Since B(#2 — x/éi 5%) % (#373137). _m71 , (m+1)(1)—(a:—1)(1)_ 2 y'a:+i y (a:+1)2 (x+1)2 then its slope is 2/(a + 1)2. But if the tangent is parallel to a: e 2y 2 2. . If the tangent intersects the curve when w = a, that is, y : in: — 1. then its slope is %. Thus. ——-—- i :> 1 (HI)2 2 (a+1)2:4 => a+1::l:2 => a:1ori3.Whena:1.y:0 andtheequationofthetangentisy—0=fizz—Dory:%m—%. When a : —3. y : 2 and the equation of the tangent is y g 2 : are + 3) 1 7 ory:§$+§. ...
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