DISP Mathematics (Scie 1500R)

MAPLE Lab. Tutorial 8 Review and Preview Monday March 13



This week we will review some of the earlier tutorials and then look ahead to the things we will be doing in the special classes.

Remember the way we defined expressions and functions: E.g.
> F:=x$\,^{\wedge}$2 +3 or > F:=x->x$\,^{\wedge}$2 + 3
>F:= proc(x) if x<=0 then 2*x+1 else x$\,^{\wedge}$2 + 3 fi end;

Remember the commands for differentiation:
> diff(F(x),x); or > D(F)(x);

Remember the command for plotting:
> with(plots): (in case you need complicated plots)
> plot({functions or expressions}, range);

Remember the command for integration:
> int(expression, variable=range);

Remember the command from last week for solving differential equations:
> dsolve({equation, initial conditions}, variable);


Define the functions:

\begin{displaymath}G(x):= x^2 - \sin(x)\quad \mbox{and} \quad H(x):= \cos(x).\end{displaymath}

Plot $H$ and $G$ together on the interval $[-2, \pi]$.

1. Find the $x$ coordinates of the two places where these curves cross.

(a)(b) $\,$

2. Find the area of the region between these two curves.

3. Find the derivative of G(H(x))

4. Find the volume of the solid of revolution formed by spinning the region bounded by the lines $y=0$, $x=-1$, $x=0$ and the curve $y=G(x)$ about the $x$-axis.(As a decimal to 2 decimal places)

5. The mean daily temperature, $T$ in degrees C, at a certain place is given by the formula

\begin{displaymath}T(t) = 20 + 10 \cos( 2\pi (t-190)/365)\end{displaymath}

where $t$ is measured in days from Jan 1.

Suppose that the rate of growth (in length L) of an insect is proportional to temperature i.e. $dL/dt = k T(t)$. Suppose that $k=0.01$ and $L(0) = 0.1$ cm.

(a) If an insect hatches on Jan 1, how big is it after 30 days?

(b) If it hatches on June 1 (= day 151) how big is it after 30 days?

Use of the following commands:
> T := 20 + 10*cos(2*Pi*(t-190)/365);
> deq := diff(L(t),t) = k*T;
> leng:= rhs(dsolve({deq,L(0)=0.1},L(t)));
> subs(t=30,k=0.01,leng);
and modify them for part (b).

6. Taylor polynomials are polynomials that approximate a function $f$ near some particular point $(x,f(x)$. They are genralizations of the idea of a tangent line. We look for the best parabola, cubic ...to fit to a particular curve at a particular point. Maple has a built in command for finding these polynomials. We will illustrate with the sine curve.

> approxsine:=proc(n,a)
local x,p,y:

Find the Taylor polynomials of degree 1, 3, 9, 21 that approximate $\sin(x)$ by asking MAPLE to evaluate (a) approxsine(1,Pi);

(b) approxsine(3,4);

(c) approxsine(9,5);

(d) approxsine(21,10);

NB: Check some of the big numbers that appear by > 19!; etc.

Tony Thompson