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Applied calculus topic summary: non-linear functions and models |
Quadratic Functions A quadratic function is a function of the form
Its graph is called a parabola. The vertex of this parabola occurs at the point on the graph with x coordinate -b/(2a). It crosses the y-axis (y-intercept) at y = c. It crosses the x-axis (x-intercept(s)) at the solutions of the quadratic equation ax2 + bx + c = 0 (if there are any). It is symmetric around the vertical line through the vertex. If the coefficient (a) of x2 is positive, it is concave up (as in the example to the right). If a is negative, it is concave down (as in the figure below). |
Example The parabola
(x + 2)(x - 4) = 0, |
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Exponential Functions
An exponential function is a function of the form
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Example
The function f(x) = 3(2x) is an exponential function with A = 3 and b = 2. It has the following graph. The following table shows the y-coordinates of points on this graph. All you do is supply the x-coordinates and press "Compute y" |
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The Laws of Exponents
If b and c are positive, and x and y are any real numbers, then the following laws hold.
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Compound Interest
Future Value
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Example
You invest $1000 at an annual rate of 4.8% interest, compounded monthly. This means that
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The Number e
The numbers
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Continuous Compounding
The number e appears in the formula for continuous compounding: If $P is invested at an annual interest rate r compounded continuously, then the accumulated amount after t years is
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Example
If $1000 is invested at an annual interest rate of 4.8% compounded continuously, then the accumulated amount after t years is
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Logarithms
The statement
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Example
The following table lists some exponential equations and their equivalent logarithmic form.
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Logarithm Identities
The following identities hold for any positive a 1 and any positive numbers x and y.
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Relationship of the Functions f(x) = logax and g(x) = ax
If a is any positive number, then the functions f(x) = logax and g(x) = ax are inverse functions. This means that
Want to learn more about inverse functions? Go to the on-line text on inverse functions. |
Examples
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Exponential Decay and Half-Life
An exponential decay function has the form
The half-life th of a substance undergoing exponential decay is the amount of time it takes for half the original quantity to decay. The half-life does not depend on the original quantity of substance present. The decay constant k and half-life th for Q are related by
Exponential Growth and Doubling Time An exponential growth function has the form
The doubling time td of a substance undergoing exponential growth is the amount of time it takes for half the original quantity to double. The doubling time does not depend on the original quantity of substance present. The growth constant k and doubling time td for Q are related by
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Examples
Decay and Half-Life:
2. If k = 0.0123, then th(0.0123) = ln 2, so the half-life is th = (ln 2)/k = (ln 2)/0.0123 ≈ 56.35 years. Growth and Doubling Time:
2. If k = 0.0123, then td(0.0123) = ln 2, so the doubling time is td = (ln 2)/k = (ln 2)/0.0123 ≈ 56.35 years. |
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Logistic Function
A logistic function has the form
Properties of the Logistic Curve
b > 1 0 < b < 1 |
Examples
N = 50, A = 24, b = 3 gives
The following figure shows the graph of f together with the exponential approximation Logistic curve: 50/(1+24*3^(-x)) Exponential curve: 2*3^x |