Transcendental Functions

(Thomas-Finney, Section 6.1, page 456)

Problem 24

Let   for x not 0.  Find the inverse function, identifying its domain and range.  As a check, writing  the inverse function as g(x), show that both f(g(x))=x  and  g(f(x)) = x.

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>

f:=1/x^3;

>	solve(y=f,x);

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

>	finv:=subs(y=x,%[1]);

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

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

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);

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

Section 6.2, page 465, Problem 35

Find the derivative with respect to  x  of   .

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Fun with the fundamental theorem:

>	assume(x>0);

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

>	simplify(%);

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

Problem 57
Evaluate   .

>

restart;

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

Section 6.3, page 472 , Problem 8

Solve for y.
                   --------------------

>	solve(ln(1-2*y)=t,y);

Problem 37
If   , compute the derivative of  y  with respect to x.

                  --------------------

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

>	solve(diff(eqn,x),diff(y(x),x));

Problem 48
Evaluate .
                   --------------------

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

Problem 77

a) Derive the linear approximation 1+x for   at  x = 0.

b) If  x  is in the interval [0, 0.2], estimate to 5 decimal places the error in using this approximation.

c) Graph and   1+x together for . Is one curve always above the other?

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(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));

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);

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);