Solução_Calculo_Stewart_6e

F.
TX.10
CHAPTER 3 REVIEW ¤ 129
(b) From part (a), a solution of y 00 =9y is y(x) =A sinh 3x + B cosh 3x. So−4 =y(0) = A sinh 0 + B cosh 0 = B,so
B = −4. Nowy 0 (x) =3A cosh 3x − 12 sinh 3x ⇒ 6=y 0 (0) = 3A ⇒ A =2,soy =2sinh3x − 4cosh3x.
55. The tangent to y =coshx has slope 1 when y 0 =sinhx =1 ⇒ x =sinh −1 1=ln 1+ √ 2 ,byEquation3.
Since sinh x =1and y =coshx = 1+sinh 2 x,wehavecosh x = √ 2. The point is ln 1+ √ 2 , √ 2 .
57. If ae x + be −x = α cosh(x + β) [or α sinh(x + β)], then
ae x + be −x = α 2 e x+β ± e −x−β
= α 2 e x e β ± e −x e −β = α
eβ e x ± α
e−β e −x . Comparing coefficients of e x
2 2
and e −x ,wehavea = α 2 eβ (1) and b = ± α 2 e−β (2). Weneedtofind α and β. Dividing equation (1) by equation (2)
gives us a b = ±e2β ⇒ () 2β =ln ± a b
⇒ β = 1 2 ln ± a b
. Solving equations (1) and (2) for e β gives us
e β = 2a α and eβ = ± α 2b ,so 2a α = ± α 2b
⇒ α 2 = ±4ab ⇒ α =2 √ ±ab.
() If a b > 0, weusethe+ sign and obtain a cosh function, whereas if a b
< 0, weusethe− sign and obtain a sinh
function.
In summary, if a and b havethesamesign,wehaveae x + be −x =2 √ ab cosh x + 1 2 ln a b
, whereas, if a and b have the
opposite sign, then ae x + be −x =2 √ −ab sinh x + 1 2 ln − a b
.
3 Review
1. (a) The Power Rule: If n is any real number, then d
dx (xn )=nx n−1 . The derivative of a variable base raised to a constant
power is the power times the base raised to the power minus one.
(b) The Constant Multiple Rule: If c is a constant and f is a differentiable function, then d
dx [cf(x)] = c d
dx f(x).
The derivative of a constant times a function is the constant times the derivative of the function.
(c) The Sum Rule: If f and g are both differentiable, then d
dx [f(x)+g(x)] = d
dx f(x)+ d g(x). The derivative of a sum
dx
of functions is the sum of the derivatives.
(d) The Difference Rule: If f and g are both differentiable, then d
d
[f(x) − g(x)] =
dx dx f(x) − d g(x). The derivative of a
dx
difference of functions is the difference of the derivatives.
(e) The Product Rule: If f and g are both differentiable, then d
d
[f(x) g(x)] = f(x)
dx dx g(x)+g(x) d f(x). The
dx
derivative of a product of two functions is the first function times the derivative of the second function plus the second
function times the derivative of the first function.