Math 156 Sp '08

Lab 2

Derivatives in Maple

If f is a function of x then

diff(f(x),x) is the derivative applied to an expressin and producing an expression

D(f) is the derivative applied to a function and producing a function.

Example: Use Maple to find the critical points and inflection points for f(x) = x^3 +12*x^2 - 3*x +15

f:= x->x^3 + 12*x^2 -3*x +15;

Zio2I0kieEc2IkYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLCoqJDkkIiIkIiIiKiRGKyIiIyIjN0YrISIkIiM6Ri1GJUYlRiU=

plot(f(x),x);

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

We can get a better view by switching the domain

plot(f(x),x=-20..6);

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

There are critical points around -8 and around 0 and an inflection point around 4. Let's get the exact values

with(student);

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

df:=diff(f(x),x);

LCgqJEkieEc2IiIiIyIiJEYkIiNDISIkIiIi

fsolve(df=0,x);

NiQkIStFYzVCIikhIiokIitjaTBKNyEjNQ==

Those are the critical points.

ddf:=diff(df,x);

LCZJInhHNiIiIiciI0MiIiI=

The second derivative can also be computed by diff(f(x),x$2);

diff(f(x),x$2);

LCZJInhHNiIiIiciI0MiIiI=

fsolve(ddf=0,x);

JCEiJSIiIQ==

This is the inflection point.

Assignment 1 Let g(x)=x^4-6x^3+1.

1A) Plot the graph of g(x) choosing the domain so that one can see all the local minima.

1B) Find the critical points of g(x) by computing the derivative of g(x) and solve the equation g'(x)=0.

1C) Find the minimum value of g(x) by checking the values of g at the critical points.

1D) Find all inflection points of g(x) by solving the equation g''(x)=0. Check on the plotted graph that this coincides with the places that the graph changes from concave upwards to concave

downwards (or vice versa).

Riemann Sums

Example: Use Maple to study the Riemann sums for f(x) = x^2 on the interval [1,3]

restart;

with(student);

The relevant Maple commands are leftbox, rightbox, leftsum and righsum. Leftbox and rightbox show the sums graphically. Leftsum and rightsum find the sums.

For each you must specify

1) the expression

2) the domain x=a..b

3) the number of subintervals

The syntax is leftbox( expression, domain, number of intervals)

f:= x->x^2;

We can see the Riemann boxes using left endpoints and 10 intervals.

Zio2I0kieEc2IkYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlKiQ5JCIiI0YlRiVGJQ==

leftbox(f(x),x=1..3,10);

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

It is clear that the area of the Riemann boxes underestimates the area under the curve.

Next we see the boxes using right hand endpoints.

rightbox(f(x),x=1..3,10);

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

It is clear that this time the boxes overestimate the area.

Let's now do the same example with shorter subintervals. This time we use n=50

leftbox(f(x),x=1..3,50);

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rightbox(f(x),x=1..3,50);

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It is nice to see both boxes at one time. For this we give the pictures names and use the display command, which needs with(plots)

left:=leftbox(f(x),x=1..3,10):right:=rightbox(f(x),x=1..3,10):

with(plots);

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

display(left,right);

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Do it again with more subintervals.

left2:=leftbox(f(x),x=1..3,50):right2:=rightbox(f(x),x=1..3,50):

display(left2,right2);

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You can see that the boxes are now better estimates.

Let's do the numbers for Example 1

L10:=leftsum(f(x),x=1..3,10): evalf(L10);

JCIrKysrISl5ISIq

R10:=rightsum(f(x),x=1..3,10): evalf(R10);

JCIrKysrIVsqISIq

Since f(x) is increasing on [1,3] the true answer is between L10 and R10. This is not yet a very accurate approximation. We will try again with larger n.

L1000:=leftsum(f(x),x=1..3,1000): evalf(L1000);

JCIrKyFvJ2UnKSEiKg==

R1000:=rightsum(f(x),x=1..3,1000): evalf(R1000);

JCIrKyFvWW4pISIq

To get accuracy of four decimal places we would have to use n= 160,000. This is a slow way to make accurate approximations.

JSFH

Assignment Part 2 Consider the function f(x) = 1/x.

2A) Graph the right and left hand sums for the interval [1,2] with n=2,4,8,16,32 subintervals.

2B) Compute the left and right hand sums at those intervals.

2C) How accurate are the estimates you get with 32 subintervals?

Assignment Part 3 Repeat 2 with g(x) = x^4 on [-1,2].