Solução_Calculo_Stewart_6e

F.
TX.10
39. We need to minimize the distance from (0, 0) to an arbitrary point (x, y) on the
SECTION 4.9 ANTIDERIVATIVES ¤ 209
curve y =(x − 1) 2 . d = x 2 + y 2
⇒
d(x) = x 2 +[(x − 1) 2 ] 2 = x 2 +(x − 1) 4 .Whend 0 =0, d will be
minimized and equivalently, s = d 2 will be minimized, so we will use Newton’s
method with f = s 0 and f 0 = s 00 .
f(x) =2x +4(x − 1) 3 ⇒ f 0 (x) = 2 + 12(x − 1) 2 ,sox n+1 = x n −
2xn +4(xn − 1)3
.Tryx1 =0.5
2+12(x n − 1) ⇒
2
x 2 =0.4, x 3 ≈ 0.410127, x 4 ≈ 0.410245 ≈ x 5 .Nowd(0.410245) ≈ 0.537841 is the minimum distance and the point on
theparabolais(0.410245, 0.347810), correct to six decimal places.
41. In this case, A =18,000, R =375,andn =5(12)=60. So the formula A = R i [1 − (1 + i)−n ] becomes
18,000 = 375
x [1 − (1 + x)−60 ] ⇔ 48x =1− (1 + x) −60 [multiplyeachtermby(1 + x) 60 ] ⇔
48x(1 + x) 60 − (1 + x) 60 +1=0. Let the LHS be called f(x),sothat
f 0 (x)=48x(60)(1 + x) 59 +48(1+x) 60 − 60(1 + x) 59
=12(1+x) 59 [4x(60) + 4(1 + x) − 5] = 12(1 + x) 59 (244x − 1)
x n+1 = x n − 48x n(1 + x n ) 60 − (1 + x n ) 60 +1
. An interest rate of 1% permonthseemslikeareasonableestimatefor
12(1 + x n ) 59 (244x n − 1)
x = i. Soletx 1 =1%=0.01,andwegetx 2 ≈ 0.0082202, x 3 ≈ 0.0076802, x 4 ≈ 0.0076291, x 5 ≈ 0.0076286 ≈ x 6 .
Thus, the dealer is charging a monthly interest rate of 0.76286% (or 9.55% per year, compounded monthly).
4.9 Antiderivatives
1. f(x) =x − 3=x 1 − 3 ⇒ F (x) = x1+1
1+1 − 3x + C = 1 2 x2 − 3x + C
Check: F 0 (x) = 1 (2x) − 3+0=x − 3=f(x)
2
3. f(x) = 1 2 + 3 4 x2 − 4 5 x3 ⇒ F (x) = 1 2 x + 3 4
Check: F 0 (x) = 1 2 + 1 4 (3x2 ) − 1 5 (4x3 )+0= 1 2 + 3 4 x2 − 4 5 x3 = f(x)
x 2+1
2+1 − 4 x 3+1
5 3+1 + C = 1 x + 1 2 4 x3 − 1 5 x4 + C
5. f(x) =(x + 1)(2x − 1) = 2x 2 + x − 1 ⇒ F (x) =2 1
3 x3 + 1 2 x2 − x + C = 2 3 x3 + 1 2 x2 − x + C
7. f(x) =5x 1/4 − 7x 3/4 ⇒ F (x) =5 x1/4+1
+1 − 7 x3/4+1
3
+ C =5x5/4
+1 5/4 − 7x7/4
7/4 + C =4x5/4 − 4x 7/4 + C
9. f(x) =6 √ x − 6√ x =6x 1/2 − x 1/6 ⇒
F (x) =6 x1/2+1
+1 − x1/6+1
1
+ C =6x3/2
+1 3/2 − x7/6
7/6 + C =4x3/2 − 6 7 x7/6 + C
1
2
6
1
4
4