Maple Solutions for Calculus

CalcLabs with Maple V

Solutions: Jacobowitz, Assignment 1

Page 10 (7,9,20)

7. Simplify the following expression:

> (2*x^2)/(x^3-1) + (3*x)/(x^2-1); simplify(``);

$2*x^2/(x^3-1)+3*x/(x^2-1)$

> plot(y,x= -100..100,-5000..5000);

$[MaplePlot]$

The discontinuity of the graph near x=1 is no longer apparent because the magnitude of the xrange is much greater than the width of the discontinuity.

Therefore the discontinuity is not visible.

> ln(a*b/c);expand(``);

$ln(a*b/c)$

Page 31 (7,8,13,17,18)

7. Plot the following expression and evaluate its limit.

> f:= (x^5 + 1) / (x^2 - 1);

$f:= (x^5+1)/(x^2-1)$

> plot(f, x = -2..0);

$[MaplePlot]$

> Limit(f,x=-1);limit(f,x = -1);

$Limit((x^5+1)/(x^2-1),x= -1)$

$-5/2$

8. Plot the following function and evaluate its limit.

> f:= x -> (x^4 - 16)/(x^2 - 4);

$f:= proc (x) options operator, arrow; (x^4-16)/(x^...$

> plot(f(x), x= 1..3);

$[MaplePlot]$

> Limit(f(x),x=2);limit(f(x),x=2);

$Limit((x^4-16)/(x^2-4),x= 2)$

$8$

13. Perform the following steps on the given expression.

> f:= sqrt((1/3)*x^2 *sin(x + Pi/6));

$f:= 1/3*sqrt(3)*sqrt(x^2*sin(x+1/6*Pi))$

> plot (f,x=0..4, 0..1); Note: The x and y ranges were choosen to clarify the graph for the first connected segment right of x=0.

If your graph looks different, this is fine becuase your x and y ranges may be different.

$[MaplePlot]$

The domain of this function is 0 to infinity.

> Digits := 11;

$Digits:= 11$

> fsolve( sin(x + Pi/6)=0,x= 0..3);

$2.6179938780$

Using the answer from fsolve as an approximation, we see that the endpoints of the first connected segment are (0,0) and (0, 2.6179938780)

17. Find decimal approximations for all roots of the equation below.

> f:= 1/x^4 - 3 + x^2;

$f:= 1/(x^4)-3+x^2$

> plot(f,x=-10..2, 0..1000);

$[MaplePlot]$

> fsolve(f=0,x,x=-1..0);fsolve(f=0,x,x=-150..0);fsolve(f=0,x,x=0..1);fsolve(f=0,x,x=0..150);

$p:= proc (x) options operator, arrow; a*x^3+b*x^2+...$

$-1.6968751402$

$.80790076412$

$1.6968751402$

18. Find the equation of the cubic that passes through points (1,2.4), (3,5.6),(4, -2.7),(7,4.7).

> p:=x -> a*x^3 + b*x^2 + c*x + d;

$eqn1:= a+b+c+d = 2.4$

> eqn1:=p(1)=2.4; eqn2:=p(3)=5.6; eqn3:=p(4)=-2.7; eqn4:=p(7)=4.7;

$eqn2:= 27*a+9*b+3*c+d = 5.6$

$eqn3:= 64*a+16*b+4*c+d = -2.7$

$eqn4:= 343*a+49*b+7*c+d = 4.7$

> soln:=solve({eqn1,eqn2,eqn3,eqn4},{a,b,c,d});

$soln:= {c = 33.773611111, d = -21.083333333, b = -...$

> f:=subs(soln,p(x));

$f:= .99861111111*x^3-11.288888889*x^2+33.773611111...$

> plot(f, x=0..10);

$[MaplePlot]$

Solution to book problem is pending

However, here are some hints:

1.) To display a limit use the Limit command. To find the numeric value of the limit use the limit command.

2.) The arguments (or parameters) of these commands are:

limit(f,x=a,dir) and Limit(f,x=a,dir) where f is an expression
x is the variable involved in the limit calculation
a is value specified in the limit calculation
dir is a direction (left, right). Use 'left' when x->a from left (-), use 'right' when x->a from right (+)