Download Math 111 – Calculus I

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Math 111 – Calculus I.
Week Number One Notes
Fall 2003
What is calculus?
Some problems that motivated the development of the calculus
1. The Area Problem
2. The Tangent Line Problem
3. The Velocity Problem
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4. Limiting Value of a Sequence of Numbers (if such a limit exists)
5. Limiting Value of a Sequence of Numeric Sums (if such a limit exists)
Review of Basics on Functions
What is a function?
Important Terminology/Properties/Characteristics of Functions
(i)
Domains/Co-Domains/Ranges of Functions
(ii)
Is a function periodic on a given domain?
(iii)
Is a function invertible on a given domain?
(iv)
Symmetries of a given function f (on a given domain D)
(a) with respect to the y-axis (f is called an even function)
(b) with respect to the origin (f is called an odd function)
(v)
Translations/Shifts/Reflections of functions through axes
(vi)
Operations on functions (pointwise addition, multiplication, composition of
functions)
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Common Classes of Functions We Will Study
1.
Polynomials
2.
Rational Functions (ratios of polynomials)
3.
Trigonometric Functions
(sin(x), cos(x), tan(x), cot(x), sec(x), csc(x))
4.
Exponential Functions (functions of the form f(x) = ax
where a is a positive constant (a is also not 1)
5.
Logarithmic Functions (functions of the form f(x) =
loga(x) where a > 0)
Example 1.1: Consider the following functions.
(a) f(x) = 6(x – 1)2 + 3
(b) g(x) = 7 cot(x)
(c) h(x) = 8x
For each of these functions, answer the following questions.
(a) What is the domain of the function (as a maximal subset of the real numbers)?
What is the range of the function?
(b) Is the function periodic on the domain specified above?
(c) Is the function invertible on the domain specified above?
(d) Is the function symmetric with respect to the y-axis? Is the function symmetric
with respect to the origin? Are there other symmetries you can determine?
(e) Using your graphing calculator, sketch the graph in an “appropriate window”.
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Example 1.2: Consider the graphs of the following two functions below.
(a) f(x)
(b) g(x)
Sketch
(i)
2f(x – 6)
(ii)
g-1(x) + 10
Example 1.3: Selected applications of functions that describe phenomena from
Newtonian physics (problem 1) and chemistry (problem 2) respectively.
(1) An application of a quadratic function
A rocket is shot straight upward at an initial velocity of 550 ft/sec. Assuming negligible
air resistance on the object, its height at time t (in seconds) is defined by the following
equation.
h(t) = -16t2 + 500t
(a) What is the maximum height above the ground that the projectile will reach?
(b) How long (in seconds) will it take for the object to reach the ground?
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(2)
An application of exponential functions – exponential growth/decay models
(radioactive decay problem)
(p. 64 of the textbook – problem number 24): An isotope of sodium, 24Na, has a half-life
of 15 hours. A sample of this isotope has an initial mass of 2 grams.
(a) Assuming a general exponential decay model for the mass (A(t)) remaining at any
time t (in hours), find a specific equation for A(t) in this case.
(b) How much isotope will be remaining after 60 hours? after 2 weeks?
Non Hand-In Homework Problems Associated with Week #1 Notes
Sections 1.5 and 1.6
Section 1.5: Problems 3,6, 7-12, 13, 15, 17, 18, 22, 23
Section 1.6: Problems 1, 3-12, 20-22, 24, 25, 27, 35-40, 43, 47-54, 57, 59
Read Sections 1.7 and 2.1 of the textbook