Download MATH 1101 Mathematical Modeling

MATH 1101 Mathematical Modeling
Review for Test #2 (1.4, Chapter 2)
1.4 Linear Functions and Linear Models
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2.1 Exponential Functions and Models
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Suggested Problems 1, 3, 5, 7, 11, 17, 19, 21, 23, 25
A quadratic function is a function having formula description of the form q(x) = ax2 + bx + c where a, b, and c are real
numbers and a ≠ 0. < Also can be written in form q(x) = a (x-h) 2 + k where (h,k) is the vertex, Note: h = -b/(2a) >
Parabolic shape (u-shape), concave up (u-shape opens up) for a > 0, concave down (u-shape opens down) for a < 0
If horizontal coordinates equally spaced, the data is exactly quadratic when second differences are (nonzero) constant.
A cubic function is a function having formula description of the form C(x) = ax3 + bx2 + cx + d where a, b, c and d are
real numbers and a ≠ 0.
< Note that cubic functions have one point of inflection>
If a > 0, end behavior of C is the same as y = x3, while if a < 0, the end behavior of C is the same as y = - x3.
If horizontal coordinates equally spaced, the data is exactly cubic when third differences are (nonzero) constant.
2.5 Choosing a Function to Fit Data
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Suggested Problems 7, 9, 11, 15, 19, 23, 25
A logistic function is a function having formula description of the form Z(x) = L/(1 + Ae-Bx) where A is a positive
real number and B is a real number satisfying B ≠ 0
y = L and y = 0 are horizontal asymptotes; (0, L/(1+A) ) is the vertical intercept
o As A approaches 0, the vertical intercept approaches (0, L)
o As A approaches infinity, the vertical intercept approaches (0, 0)
The function either increases (B>0) OR decreases (B<0), as |B| approaches infinity, the graph is steeper in the middle.
One inflection point (a point around which concavity changes)
If data looks logistic but neither end approaches zero, a vertical shift of the data may result in a better fit.
Alignment may be necessary if not all horizontal coordinates are close to zero.
2.4 Quadratic and Cubic Functions and Models
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Suggested Problems 1, 2, 3, 4, 5, 7, 9, 11, 13
A logarithmic function is a function having formula description of the form l(x) = a + b ln(x) where b is a real number
satisfying b ≠ 0, and a is a real number.
(e-a/b, 0) is the horizontal intercept, no horizontal asymptotes, the vertical axis is a vertical asymptote
Increases without bound (b>0), or decreases without bound (b<0) but does so very slowly
No inflection points: Always concave down (b>0) OR Always concave up (b<0)
Logarithmic functions are only defined for positive real numbers, the vertical axis is a vertical asymptote
Alignment (horizontal shift of the data) may be necessary if not all horizontal coordinates are positive
2.3 Logistic Functions and Models
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Suggested Problems 1, 3, 9, 11, 13, 15, 17, 21, 27
An exponential function is a function having formula description of the form E(x) = bax where b is a real number and
a is a real number satisfying a >0 and a ≠ 1. [Note: in most models, we'll find b > 0]
Exponential functions change very fast: the rate of increase (r > 0) or decrease (-1 < r < 0) is ultimately infinite.
Exponential functions are characterized by a constant percentage rate of change, i.e. for each one unit increase in the
horizontal coordinate, the vertical coordinate of an exponential function changes by the same percentage.
If (x, y) and (r, s) are points on an exponential function E(x) = ba x, then y/s = axr
When horizontal coordinates are equally spaced, the date is exactly exponential if and only if first ratios are constant.
E(x) = bax = b(1+r) x: if r > 0 (a >1) then the growth factor is 1+r, percentage growth rate is 100r percent
E(x) = bax = b(1+r) x: if -1 < r < 0 (a >1) then the decay factor is 1+r, percentage decay rate is 100|r| percent
For E(x) = bax , (0, b) is the vertical intercept, the horizontal axis is a horizontal asymptote; no vertical asymptotes
If data looks exponential but neither end approaches zero, a vertical shift of the data may result in a better fit
2,2 Logarithmic Functions and Models
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Suggested Problems 1, 3, 5, 7, 9, 13, 21, 23, 25, 29
A linear function is a function having formula description of the form L(x) = mx + b where m and b are real numbers.
The slope is the signed vertical change corresponding to a one unit increase in the horizontal coordinate
Using similar triangles, we can show that every pair of points on a given line determine the slope (y2-y1)/(x2-x1) = m
When horizontal coordinates are equally spaced, the data is exactly linear if and only if first differences are constant.
Point-slope form of the equation of a line (y - y1) = m(x - x1) where m is the slope and (x1, y1) is a point on the line
Slope-intercept form of the equation of a line y = L(x) = mx + b where m is the slope and (0,b) is the vertical intercept
Linear Models for sets of points which are approximately linear
o Linear Regression (also known as least squares regression: minimizing the sum of the squared vertical errors)
o Interpolation, Extrapolation
o Four elements of a model (a linear function, a description including units of both input and output, domain)
Suggested Problems 1, 2, 3, 4, 5, 6, 7, 9, 11, 13, 15
Shape of scatter plot may focus our attention on particular models, anticipated end behavior should also be a factor
Linear and Exponential are characterized, for Quadratic and Cubic there are characterizations for equally spaced data