MATH 2253 - Graphing with Calculus and Calculator: Worksheet 1
Joel Fowler
The nature of the problems here is such that the TI-89 is virtually essential for many of the computations. At a
minimum you should find the machine useful for computing derivatives, solving equations, evaluating functions,
simplifying expressions, and sketching graphs. Note that you must use calculus theorems to justify your
answers. For example, to argue that a function has a relative maximum at x, you can use your calculator to
find f’, you can graph f’, and based on the graph together with the complete set of solutions to the equation f’
(x)=0, you can argue that f’ is positive to the left of x, negative to the right of x, and zero at x, thus, f has a
relative maximum at x. It is not acceptable to give unsupported answers using calculator functions instead of
calculus. Use the machine to find derivatives, when necessary, and to solve equations. Use the graphing
feature to help you solve inequalities, for example, f’ (x)>0.
All points and values requested are to be exact with no decimal approximations, unless otherwise requested.
1. Let 43 2
f (x) = 150 - 7600 + 95997 + 479988x + 20
xx x .
a) Find all critical values and their y-coordinates for this function. Identify each point as a relative maximum, a
relative minimum, or neither.
b) Find the absolute minimum value of this function over its domain.
c) Find the x coordinates of all inflection points.
2. Let 2
2
- 50x + 30
x
f (x) = + 100
x.
a) Find the points at which this function takes on an absolute maximum and an absolute minimum value.
b) Find the x-coordinates, to three decimal places, of all inflection points.
3. Let 2
63
f (x) = (x + 30)
x
a) Find all critical values and their y-coordinates for this function. Identify each point as a relative maximum, a
relative minimum, or neither.
b) Find the x-coordinates of all inflection points.
4. Find the points at which
4
3
2
x
f (x) = + 2x + 5
x have absolute maximum values, and absolute minimum
values.
