Mattingly "AI & Prompt Design: Large Language Models"
Lesson 2: A Catalog of Essential Functions
1. Section 1.2
A catalog of essential functions
V63.0121.006/016, Calculus I
January 21, 2010
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. . . . . .
2. Outline
Modeling
Classes of Functions
Linear functions
Other Polynomial functions
Other power functions
Rational functions
Trigonometric Functions
Exponential and Logarithmic functions
Transformations of Functions
Compositions of Functions
. . . . . .
3. The Modeling Process
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Real-world
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. m
. odel Mathematical
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Problems Model
s
. olve
.est
t
. i
.nterpret .
Real-world
. Mathematical
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Predictions Conclusions
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5. The Modeling Process
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Real-world
.
. m
. odel Mathematical
.
Problems Model
s
. olve
.est
t
. i
.nterpret .
Real-world
. Mathematical
.
Predictions Conclusions
S
. hadows F
. orms
. . . . . .
6. Outline
Modeling
Classes of Functions
Linear functions
Other Polynomial functions
Other power functions
Rational functions
Trigonometric Functions
Exponential and Logarithmic functions
Transformations of Functions
Compositions of Functions
. . . . . .
7. Classes of Functions
linear functions, defined by slope an intercept, point and
point, or point and slope.
quadratic functions, cubic functions, power functions,
polynomials
rational functions
trigonometric functions
exponential/logarithmic functions
. . . . . .
8. Linear functions
Linear functions have a constant rate of growth and are of the
form
f(x) = mx + b.
. . . . . .
9. Linear functions
Linear functions have a constant rate of growth and are of the
form
f(x) = mx + b.
Example
In New York City taxis cost $2.50 to get in and $0.40 per 1/5
mile. Write the fare f(x) as a function of distance x traveled.
. . . . . .
10. Linear functions
Linear functions have a constant rate of growth and are of the
form
f(x) = mx + b.
Example
In New York City taxis cost $2.50 to get in and $0.40 per 1/5
mile. Write the fare f(x) as a function of distance x traveled.
Answer
If x is in miles and f(x) in dollars,
f(x) = 2.5 + 2x
. . . . . .
13. Solution
The point-slope form of the equation for a line is appropriate
here: If a line passes through (x0 , y0 ) with slope m, then the
line has equation
y − y0 = m(x − x0 )
. . . . . .
14. Solution
The point-slope form of the equation for a line is appropriate
here: If a line passes through (x0 , y0 ) with slope m, then the
line has equation
y − y0 = m(x − x0 )
80 − 70 10 1
The slope of our line is = =
173 − 113 60 6
. . . . . .
15. Solution
The point-slope form of the equation for a line is appropriate
here: If a line passes through (x0 , y0 ) with slope m, then the
line has equation
y − y0 = m(x − x0 )
80 − 70 10 1
The slope of our line is = =
173 − 113 60 6
So an equation for T and N is
1 1 113
T − 70 = (N − 113) =⇒ T = N − + 70
6 6 6
. . . . . .
16. Solution
The point-slope form of the equation for a line is appropriate
here: If a line passes through (x0 , y0 ) with slope m, then the
line has equation
y − y0 = m(x − x0 )
80 − 70 10 1
The slope of our line is = =
173 − 113 60 6
So an equation for T and N is
1 1 113
T − 70 = (N − 113) =⇒ T = N − + 70
6 6 6
37
If N = 150, then T = + 70 = 76 1 ◦ F
6
6
. . . . . .
17. Other Polynomial functions
Quadratic functions take the form
f(x) = ax2 + bx + c
The graph is a parabola which opens upward if a > 0,
downward if a < 0.
. . . . . .
18. Other Polynomial functions
Quadratic functions take the form
f(x) = ax2 + bx + c
The graph is a parabola which opens upward if a > 0,
downward if a < 0.
Cubic functions take the form
f(x) = ax3 + bx2 + cx + d
. . . . . .
20. Example
A parabola passes through (0, 3), (3, 0), and (2, −1). What is the
equation of the parabola?
Solution
The general equation is y = ax2 + bx + c.
. . . . . .
21. Example
A parabola passes through (0, 3), (3, 0), and (2, −1). What is the
equation of the parabola?
Solution
The general equation is y = ax2 + bx + c. Each point gives an
equation relating a, b, and c:
3 = a · 02 + b · 0 + c
−1 = a · 2 2 + b · 2 + c
0 = a · 32 + b · 3 + c
. . . . . .
22. Example
A parabola passes through (0, 3), (3, 0), and (2, −1). What is the
equation of the parabola?
Solution
The general equation is y = ax2 + bx + c. Each point gives an
equation relating a, b, and c:
3 = a · 02 + b · 0 + c
−1 = a · 2 2 + b · 2 + c
0 = a · 32 + b · 3 + c
Right away we see c = 3. The other two equations become
−4 = 4a + 2b
−3 = 9a + 3b
. . . . . .
27. Other power functions
Whole number powers: f(x) = xn .
1
negative powers are reciprocals: x−3 = 3 .
x
√
fractional powers are roots: x1/3 = 3 x.
. . . . . .
28. Rational functions
Definition
A rational function is a quotient of polynomials.
Example
x 3 (x + 3 )
The function f(x) = is rational.
(x + 2)(x − 1)
. . . . . .
29. Trigonometric Functions
Sine and cosine
Tangent and cotangent
Secant and cosecant
. . . . . .
30. Exponential and Logarithmic functions
exponential functions (for example f(x) = 2x )
logarithmic functions are their inverses (for example
f(x) = log2 (x))
. . . . . .
31. Outline
Modeling
Classes of Functions
Linear functions
Other Polynomial functions
Other power functions
Rational functions
Trigonometric Functions
Exponential and Logarithmic functions
Transformations of Functions
Compositions of Functions
. . . . . .
32. Transformations of Functions
Take the sine function and graph these transformations:
( π)
sin x +
( 2
π)
sin x −
2
π
sin (x) +
2
π
sin (x) −
2
. . . . . .
33. Transformations of Functions
Take the sine function and graph these transformations:
( π)
sin x +
( 2
π)
sin x −
2
π
sin (x) +
2
π
sin (x) −
2
Observe that if the fiddling occurs within the function, a
transformation is applied on the x-axis. After the function, to the
y-axis.
. . . . . .
34. Vertical and Horizontal Shifts
Suppose c > 0. To obtain the graph of
y = f(x) + c, shift the graph of y = f(x) a distance c units
y = f(x) − c, shift the graph of y = f(x) a distance c units
y = f(x − c), shift the graph of y = f(x) a distance c units
y = f(x + c), shift the graph of y = f(x) a distance c units
. . . . . .
35. Vertical and Horizontal Shifts
Suppose c > 0. To obtain the graph of
y = f(x) + c, shift the graph of y = f(x) a distance c units
upward
y = f(x) − c, shift the graph of y = f(x) a distance c units
y = f(x − c), shift the graph of y = f(x) a distance c units
y = f(x + c), shift the graph of y = f(x) a distance c units
. . . . . .
36. Vertical and Horizontal Shifts
Suppose c > 0. To obtain the graph of
y = f(x) + c, shift the graph of y = f(x) a distance c units
upward
y = f(x) − c, shift the graph of y = f(x) a distance c units
downward
y = f(x − c), shift the graph of y = f(x) a distance c units
y = f(x + c), shift the graph of y = f(x) a distance c units
. . . . . .
37. Vertical and Horizontal Shifts
Suppose c > 0. To obtain the graph of
y = f(x) + c, shift the graph of y = f(x) a distance c units
upward
y = f(x) − c, shift the graph of y = f(x) a distance c units
downward
y = f(x − c), shift the graph of y = f(x) a distance c units to
the right
y = f(x + c), shift the graph of y = f(x) a distance c units
. . . . . .
38. Vertical and Horizontal Shifts
Suppose c > 0. To obtain the graph of
y = f(x) + c, shift the graph of y = f(x) a distance c units
upward
y = f(x) − c, shift the graph of y = f(x) a distance c units
downward
y = f(x − c), shift the graph of y = f(x) a distance c units to
the right
y = f(x + c), shift the graph of y = f(x) a distance c units to
the left
. . . . . .
39. Outline
Modeling
Classes of Functions
Linear functions
Other Polynomial functions
Other power functions
Rational functions
Trigonometric Functions
Exponential and Logarithmic functions
Transformations of Functions
Compositions of Functions
. . . . . .
41. Composing
Example
Let f(x) = x2 and g(x) = sin x. Compute f ◦ g and g ◦ f.
. . . . . .
42. Composing
Example
Let f(x) = x2 and g(x) = sin x. Compute f ◦ g and g ◦ f.
Solution
f ◦ g(x) = sin2 x while g ◦ f(x) = sin(x2 ). Note they are not the
same.
. . . . . .
43. Decomposing
Example
√
Express x2 − 4 as a composition of two functions. What is its
domain?
Solution √
We can write the expression as f ◦ g, where f(u) = u and
g(x) = x2 − 4. The range of g needs to be within the domain of f.
To insure that x2 − 4 ≥ 0, we must have x ≤ −2 or x ≥ 2.
. . . . . .