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Applied calculus topic summary: nonlinear 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 yaxis (yintercept) at y = c. It crosses the xaxis (xintercept(s)) at the solutions of the quadratic equation ax^{2} + bx + c = 0 (if there are any). It is symmetric around the vertical line through the vertex. If the coefficient (a) of x^{2} 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, 

Exponential Functions
An exponential function is a function of the form

Example
The function f(x) = 3(2^{x}) is an exponential function with A = 3 and b = 2. It has the following graph. The following table shows the ycoordinates of points on this graph. All you do is supply the xcoordinates and press "Compute y" 

The Laws of Exponents
If b and c are positive, and x and y are any real numbers, then the following laws hold.


Compound Interest
Future Value

Example
You invest $1000 at an annual rate of 4.8% interest, compounded monthly. This means that


The Number e
The numbers


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

Example
If $1000 is invested at an annual interest rate of 4.8% compounded continuously, then the accumulated amount after t years is


Logarithms
The statement

Example
The following table lists some exponential equations and their equivalent logarithmic form.


Logarithm Identities
The following identities hold for any positive a 1 and any positive numbers x and y.


Relationship of the Functions f(x) = log_{a}x and g(x) = a^{x}
If a is any positive number, then the functions f(x) = log_{a}x and g(x) = a^{x} are inverse functions. This means that
Want to learn more about inverse functions? Go to the online text on inverse functions. 
Examples


Exponential Decay and HalfLife
An exponential decay function has the form
The halflife t_{h} of a substance undergoing exponential decay is the amount of time it takes for half the original quantity to decay. The halflife does not depend on the original quantity of substance present. The decay constant k and halflife t_{h} for Q are related by
Exponential Growth and Doubling Time An exponential growth function has the form
The doubling time t_{d} 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 t_{d} for Q are related by

Examples
Decay and HalfLife:
2. If k = 0.0123, then t_{h}(0.0123) = ln 2, so the halflife is t_{h} = (ln 2)/k = (ln 2)/0.0123 ≈ 56.35 years. Growth and Doubling Time:
2. If k = 0.0123, then t_{d}(0.0123) = ln 2, so the doubling time is t_{d} = (ln 2)/k = (ln 2)/0.0123 ≈ 56.35 years. 

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 