> evalf(`+`(sqrt(2)+sqrt(3), -sqrt(`+`(2, 3)))); 1
 

.910196393 

Therefore, sqrt(2)+sqrt(3)>sqrt(2+3). 

 

> x := 4.15231
 

(Typesetting:-mprintslash)([x := 4.15231], [4.15231]) 

> (x^2-tan(x))/(x^3-1)
 

.2216505788 

> unassign('x'); 1
 

You can also do problem 2 using the "subs" command as follows: 

> subs(x = 4.15231, (x^2-tan(x))/(x^3-1))
 

.2442413386-0.1416575195e-1*tan(4.15231) 

> evalf(%); 1
 

.2216505789 

 

 

> solve(exp(x^2) = 10, x)
 

ln(10)^(1/2), -ln(10)^(1/2) 

 

 

> diff(sin(a*x)/cos(b*x), x, x)
 

-sin(a*x)*a^2/cos(b*x)+2*cos(a*x)*a*sin(b*x)*b/cos(b*x)^2+2*sin(a*x)*sin(b*x)^2*b^2/cos(b*x)^3+sin(a*x)*b^2/cos(b*x) 

 

 

> int(4*sin(x)+cos(2*x)-1, x = 0 .. Pi)
 

8-Pi 

> evalf(int(4*sin(x)+cos(2*x)-1, x = 0 .. Pi))
 

4.858407346 

 

 

> plot(4*sin(x)+cos(2*x)-1, x = 0 .. Pi); 1
 

Plot 

You can customize the plot by using options at the end of the plot command.  See plot/options in the help menu for all of your options.  The plot3d command (for 3 dimensional plotting) has even more options. 

> plot(4*sin(x)+cos(2*x)-1, x = 0 .. Pi, axes = boxed, thickness = 7, color = plum, scaling = constrained, title =
plot(4*sin(x)+cos(2*x)-1, x = 0 .. Pi, axes = boxed, thickness = 7, color = plum, scaling = constrained, title =
 

Plot 

In problem 7 we are given the equation of the curve, so we'll need to use the implicitplot command.  This command is in the plots package. 

> with(plots); -1
 

> implicitplot((x^2-1)^2+(y^2-1)*(y+1)^2 = 0, x = -2 .. 2, y = -1 .. 1, color = black, scaling = constrained)
implicitplot((x^2-1)^2+(y^2-1)*(y+1)^2 = 0, x = -2 .. 2, y = -1 .. 1, color = black, scaling = constrained)
implicitplot((x^2-1)^2+(y^2-1)*(y+1)^2 = 0, x = -2 .. 2, y = -1 .. 1, color = black, scaling = constrained)
 

Plot 

This looks a little chunky at the bottom, so let's adjust the "numpoints" value (short for number of points) to see if we can correct this. 

> implicitplot((x^2-1)^2+(y^2-1)*(y+1)^2 = 0, x = -2 .. 2, y = -1 .. 1, color = black, scaling = constrained, numpoints = 50)
implicitplot((x^2-1)^2+(y^2-1)*(y+1)^2 = 0, x = -2 .. 2, y = -1 .. 1, color = black, scaling = constrained, numpoints = 50)
implicitplot((x^2-1)^2+(y^2-1)*(y+1)^2 = 0, x = -2 .. 2, y = -1 .. 1, color = black, scaling = constrained, numpoints = 50)
 

Plot 

> implicitplot((x^2-1)^2+(y^2-1)*(y+1)^2 = 0, x = -2 .. 2, y = -1 .. 1, color = black, scaling = constrained, numpoints = 10000)
implicitplot((x^2-1)^2+(y^2-1)*(y+1)^2 = 0, x = -2 .. 2, y = -1 .. 1, color = black, scaling = constrained, numpoints = 10000)
implicitplot((x^2-1)^2+(y^2-1)*(y+1)^2 = 0, x = -2 .. 2, y = -1 .. 1, color = black, scaling = constrained, numpoints = 10000)
 

Plot 

That's much smoother, but the downside is that it takes much longer to draw. 

 

> simplify((x^3-sin(x)^2-cos(x)^2)/(x-1))
 

x^2+x+1