Workshop 1
Preliminary Exercises
• Find the errors in the following calculations and rewrite the sentence so that it’s correct:
a) .
b)
c)
d)
e)
f) means or .
• Determine whether the following statements are true or false. For those which are true (if any), explain why. For those which are false (if any), provide an example showing they are false (such an example is called a counter-example).
a) / d) / h)b) / e) / i)
c) / f) / j)
• The following diagram depicts the interval [1,3).
Draw similar diagrams to illustrate the intervals specified below.
a) [2,3] / b) (2,3) / c) (2,3]d) / e) / f) [6,6]
g) (6,6) / h) (9, -5]
Functions as Models
Functions: Definition, Notation, Domain, Range
• A homeowner mows the lawn every Wednesday afternoon. Sketch a rough graph of the height of the grass as a function of time over the course of a four-week period.
• Define what we mean by the domain and range of a function. Then answer the following questions:
a) If two functions have the same domain must their graphs cross?
b) If two functions have the same range, must their graphs cross?
c) If two functions have the same domain and range, must their graphs cross?
• This question refers to the function .
a) Calculate f(2), f(3), and f(-1).
b) Calculate f(1/3), f(3/4) f(1/2).
What is the domain of f ?
Graphing functions
• An aircraft leaves Rochester for Chicago and, though it arrives on time, it is put into a holding pattern because of heavy air traffic and circles the city for an hour before receiving clearance for landing[1].
a) Suppose H(t) is the height of the aircraft at time t. Draw the graph of H.
b) Suppose R(t) is the plane's distance from Rochester. Draw the graph of R.
c) Does your graph reflect the fact that the plane speeds up, initially, travels most of the way at a constant rate, and then slows down? If not, try again. If so, what characteristic of the graph indicates this physical reality?
d) Suppose C(t) is the plane's distance from Chicago. Draw the graph of C.
e) Using your graph from part (b), how would you calculate the length of time it takes the plane to orbit Chicago one time? Would the same method work if you used your graph from part (d)?
Piecewise defined functions
• When you drive their compact car x miles, a certain rental company charges f(x),dollars, where
Describe this company’s pricing policy in plain English (no equations). Be sure to give economic interpretations of the constants 30, 0.07 and 100 that appear in the pricing formula.
• The Heaviside function is defined piecewise by .
a) Draw the graph of H.
b) Suppose that , and we set . Calculate s(-200), s(-2), s(-1/2), s(0), s(1/2), s(2), s(3), and s(5).
c) In English sentences, explain why we think of the Heaviside function as a switch.
d) Without using a graphing utility, sketch a graph of y=s(x).
• Suppose that , and H(x) is the Heaviside function, defined below. On separate axes, sketch the graphs of f, H, and H(x)f(x).
• Suppose f and g are the functions described below. Find a formula for f+g and draw its graph. Then find a formulas for fg, and gg.
Symmetry
• A function is said to be even if f(x) = f(-x), and odd if f(x)= - f(-x). That is, a function is even if f(x) and f(-x) are the same, and is odd if f(x) and f(-x) are opposites.
· Draw the graphs of an even function and an odd function.
· Draw the graph of a function that is neither even nor odd.
· Draw the graph of a function that is both even and odd.
Increasing and Decreasing Functions
• Define what it means to say that a function is decreasing. Then do the following:
a) Draw the graph of a function that is decreasing, never zero, and has f(-1) = 2.
b) Draw the graph of a function that is decreasing, never zero, and has f(1) = -2.
c) Draw the graph of a function that is decreasing, never zero, and has f(-1) = 2 and f(1) = -2.
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