Calculus & Analytic Geometry I, MTH 168
(Instructor: M. USMAN)
Objectives: In this lab we will learn to find the critical numbers and absolute maximum and minimum values of a function.
Critical Points Using Solve
Let us see few examples of finding the critical points and plotting tangents at critical points
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.)
Step 1: Use Maple's differentiation abilities to find the derivative of .
Step 2: Use Maple's algebra ability to determine what value(s) of make this derivative equal zero.
Step 3: Use the evalf command to estimate the size of the numbers you found as critical point.
Step 4: Plot the function on an interval large enough to show the behavior of at both of these critical points.
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
To find the critical numbers of in , we must compute the derivative of (call it ):
has a root at -1 and 2
Next, compute the value of at the critical numbers and the endpoints:
We see that the absolute maximum occurs at , y = f()=8 and the absolute minimum at , y = f(2)=-19.
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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