Calculus & Analytic Geometry I, MTH 168

(Instructor: M. USMAN)

Lab#4

Objectives: In this lab we will learn to find the critical numbers and absolute maximum and minimum values of a function.

 > restart;with(student):

Critical Points Using Solve

solve(equations, variables)

Let us see few examples of finding the critical points and plotting tangents at critical points

 > f:=x->2*x^3-3*x^2-5*x+6;

 (1.1)

 > derf:=diff(f(x),x);

 (1.2)

 > points:=solve(derf=0.0,x);

 (1.3)

 > showtangent(f(x), points[2],x=-6..6,y=-10..10);

 > g:=t->t^3+3*t^2-24*t;

 (1.4)

 > derg:=diff(g(t),t);

 (1.5)

 > cpnts:=solve(derg=0.0,t);

 (1.6)

 > showtangent(g(t), points[2]);

Example: Find the critical points of the function and use a graph to estimate the minimum value of .  (Recall that critical points are the x-values where the derivative is zero.)

 > f:=x->3*x^4+4*x^3-6*x^2;

 (1.7)

Step 1: Use Maple's differentiation abilities to find the derivative of .

 > dervif:=diff(f(x),x);

 (1.8)

Step 2: Use Maple's algebra ability to determine what value(s) of make this derivative equal zero.

 > solve(dervif=0,x);

 (1.9)

Step 3: Use the evalf command to estimate the size of the numbers you found as critical point.

 > evalf(%);

 (1.10)

Step 4: Plot the function on an interval large enough to show the behavior of at both of these critical points.

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

Absolute Maximum and Minimum

To find the absolute maximum and minimum of a continuous function on a closed interval, we must take its extreme values at either its critical numbers in the interval or the endpoints of the interval.  For example, on , consider the function

 > restart;

 > f:=x->2*x^3-3*x^2-12*x+1;

 (1)

To find the critical numbers of in , we must compute the derivative of (call it ):

 > fprime:=D(f);

 > plot({f,fprime},-2..3);

 > solve(fprime(x)=0,x);

 (2)

has a root at -1 and 2

Next, compute the value of at the critical numbers and the endpoints:

 > f(-2),f(-1),f(2),f(3);

We see that the absolute maximum occurs at , y = f()=8 and the absolute minimum at , y = f(2)=-19.

Homework Problems:

1) Find the critical numbers of f(θ)=2cos(θ)+(sin(θ))^2.

2) Find the critical numbers of f(x)=

3) Find the critical numbers of  f(t)=t

4) Find the absolute maximum and absolute minimum values of  f(x)= x^3-6x^2+9x+2

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