Thomas Calculus 13th [Solutions]

109 110. Example CAS commands:
Section 3.8 Derivatives of Inverse Functions and Logarithms 179
Maple:
with( plots );
eq := cos(y) = x^(1/5);
domain := 0 .. 1;
x0 := 1/2;
f := unapply( solve( eq, y ), x );
# (a)
Df := D(f);
plot( [f(x),Df(x)], x=domain, color=[red,blue], linestyle=[1,3], legend=["y=f(x)",y=f'(x)"],
title="#110(a) (Section 3.8)" );
q1 := solve( eq, x );
# (b)
g := unapply( q1, y );
m1 := Df(x0);
# (c)
t1 := f(x0)+m1*(x-x0);
y=t1;
m2 := 1/Df(x0);
# (d)
t2 := g(f(x0)) + m2*(x-f(x0));
y=t2;
domaing := map(f,domain);
# (e)
p1 := plot( [f(x),x], x=domain, color=[pink,green], linestyle=[1,9], thickness=[3,0] ):
p2 := plot( g(x), x=domaing, color=cyan, linestyle=3, thickness=4 ):
p3 := plot( t1, x=x0-1..x0+1, color=red, linestyle=4, thickness=0 ):
p4 := plot( t2, x=f(x0)-1..f(x0)+1, color=blue, linestyle=7, thickness=1 ):
p5 := plot( [ [x0,f(x0)], [f(x0),x0] ], color=green ):
display( [p1,p2,p3,p4,p5], scaling=constrained, title="#110(e) (Section 3.8)" );
Mathematica: (Assigned function and values for a, b, and x0 may vary)
For problems 109 and 110, the code is just slightly altered. At times, different parts of solutions need to be
used as in the definitions of f[x] and g[y]
Clear[x, y]
{a,b} = {0, 1}; x0 = 1/2 ;
1/5
eqn = Cos[y] == x
soly = Solve[eqn, y]
f[x_] = y /. soly[[2]]
Plot[{f[x], f'[x]}, {x, a, b}]
solx = Solve[eqn, x]
g[y_] = x /. solx[[1]]
y0 = f[x0]
ftan[x_] = y0+f'[x0] (x-x0)
gtan[y_] = x0 + 1/f'[x0] (y-y0)
Plot[{f[x], ftan[x], g[x], gtan[x], Identity[x]}, {x, a, b},
Epilog Line[{x0, y0}, {y0, x0}}], PlotRange {{a, b}, {a, b}}, AspectRatio Automatic]
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