CHAPTER 6 - Transcendental Functions

Section 6.1, page 454

Problem 24

>   f:=1/x^3;

f := 1/(x^3)

>   solve(y=f,x);

1/(y^(1/3)), -1/(2*y^(1/3))+1/2*I*3^(1/2)/y^(1/3), -1/(2*y^(1/3))-1/2*I*3^(1/2)/y^(1/3)

We'll take the first one, because it's real!

>   finv:=subs(y=x,%[1]);

finv := 1/(x^(1/3))

>   simplify(subs(x=finv,f));

x

>   simplify(subs(x=f,finv),symbolic);

x

So we're ok -- let's plot to make sure:

>   plot({f,x,finv},x=0..5,-1..5,color=blue,thickness=2,scaling=constrained);

[Maple Plot]

You can see that the two curves are reflections of one another (but it's not easy!).

Section 6.2, page 465

Problem 35

Fun with the fundamental theorem:

>   assume(x>0);

>   diff(int(ln(sqrt(t)),t=x^2/2..x^2),x);

2*x*ln(x)-x*ln(1/2*2^(1/2)*x)

>   simplify(%);

1/2*x*(2*ln(x)+ln(2))

The little tildes "~" indicate that we have made an assumption about x.

Problem 57

>   restart;

>   Int(2*ln(x)/x,x=1..2)=int(2*ln(x)/x,x=1..2);

Int(2*ln(x)/x,x = 1 .. 2) = ln(2)^2

Section 6.3, page 472

Problem 8

>   solve(ln(1-2*y)=t,y);

-1/2*exp(t)+1/2

Problem 37

>   eqn:=ln(y(x))=exp(y(x))*sin(x);

eqn := ln(y(x)) = exp(y(x))*sin(x)

>   solve(diff(eqn,x),diff(y(x),x));

-exp(y(x))*cos(x)*y(x)/(-1+exp(y(x))*sin(x)*y(x))

Problem 48

>   Int(exp(x/4),x=0..ln(16))=int(exp(x/4),x=0..ln(16));

Int(exp(1/4*x),x = 0 .. ln(16)) = 4*4^(1/2)-4

Problem 77

(a) We do the usual tangent line idiom:

>   f:=exp(x); tanline:=simplify(subs(x=0,f)+subs(x=0,diff(f,x))*(x-0));

f := exp(x)

tanline := 1+x

Let's plot first, so we can see where the error is worst:

>   plot({f,tanline},x=-0.2..0.5,color=blue,thickness=2);

[Maple Plot]

The line is always below the curve, and it's worst at x=0.2. So our upper bound for the error is

>   exp(0.2)-subs(x=0.2,tanline);

.21402758e-1