linapprx.html

Linear Approximations

In this worksheet we'll work with the linear approximation formula 2.33 of our text which says that if f is differentiable near then f(x) is approximately

Tables and graphs will be used to help show where the approximation is a good one.
This worksheet will use the Maple function option under Enter Expression of MenuMaple. Before starting be sure that you have the calculus menu of MenuMaple open and click in the first command region.

EXAMPLE ONE

Select Enter Expression and click on the function option . The dialogue box will now have three input areas, one for the name of the function, one for the name of the independent variable, and one for the expression. Enter f as the name of the function, x as the independent variable and sqrt(x) as the expression.

> f:=x->sqrt(x);

The output that you will see on the screen is a Maple's mapping notation for functions which will alllow us to use f(x) notation in Maple.

Use the differentiate button to find the derivative of f with respect to x. Call the result df.

> df:=D(f);

We will try to approximate the square root of x for x near . Recall that to get a linear
approximation to a function f for x near we use

To get a linear approximation for x near 4 use Enter Function to enter a function called Lapprox with x as the independent variable and f(4) + df(4)*(x-4) as the expression.

> Lapprox:=x->f(4)+df(4)*(x-4);

Make a table of values showing x, the actual square root, and the approximate square root found by the linear approximation. Use x between 3.8 and 4.2 in steps of 0.02. To do this use table of values with f(x) and Lapprox(x) as expressions (separated by a comma).

> with(math191):
> valutable([f(x),Lapprox(x)],x=3.8..4.2,'step'=0.02);

x funct1(x) funct2(x) ---------------------------------------------- 3.8 1.949358869 1.950000000 3.820000000 1.954482029 1.955000000 3.840000000 1.959591794 1.960000000 3.860000000 1.964688270 1.965000000 3.880000000 1.969771560 1.970000000 3.900000000 1.974841766 1.975000000 3.920000000 1.979898987 1.980000000 3.940000000 1.984943324 1.985000000 3.960000000 1.989974874 1.990000000 3.980000000 1.994993734 1.995000000 4.000000000 2.000000000 2.000000000 4.020000000 2.004993766 2.005000000 4.040000000 2.009975124 2.010000000 4.060000000 2.014944168 2.015000000 4.080000000 2.019900988 2.020000000 4.100000000 2.024845673 2.025000000 4.120000000 2.029778313 2.030000000 4.140000000 2.034698995 2.035000000 4.160000000 2.039607805 2.040000000 4.180000000 2.044504830 2.045000000 4.200000000 2.049390153 2.050000000

In the table what do you notice f(x) and Lapprox(x) when x gets close to ?

Make a graph of the original f(x) and the linear approximation, Lapprox(x), over the interval from 0 to 6.

> plot([f(x),Lapprox(x)],x=0..6,title=`actual in blue aprroximation in red`,color=[blue,red]);

What happens as x gets closer to xo=4?

EXAMPLE TWO Approximate the sine of 31 degrees.
Enter g as the name of the function, x as the independent variable, and sin(x) as the expression.

> g:=x->sin(x);

Now find its derivative with respect to x and give the result the name dg.

> dg:=D(g);

Maple expects angles in radians. Since 31 degrees is near Pi/6, we will find a linear approximation near Pi/6. Enter Lsin as the name of the function, x as the independent variable and

g(Pi/6) + dg(Pi/6)*(x - Pi/6) as the expression.

> Lsin:=x->g(Pi/6)+dg(Pi/6)*(x-Pi/6);