Main Page Everything for Finite Math Everything for Applied Calc Everything Topic Summaries On Line Tutorials On Line Utilities               ← Previous Summary Next Summary → Review Exercises E-mail Webmaster Textbook Lleveme a la Páaina Español Applied calculus topic summary: non-linear functions and models Tools: Function Evaluator & Grapher | Excel Grapher | Simple Regression Utility Subtopics: Quadratic Functions | Exponential Functions | Laws of Exponents | Compound Interest | The Number e | Continuous Compounding | Logarithms | Logarithm Identities | Relationship of Log and Exponential Functions | Half-Life and Doubling Time | Logistic Function A quadratic function is a function of the form

f(x) = ax2 + bx + c     (with a ≠ 0).

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

y = x2 - 2x - 8
has a vertex with x-coordinate
 - b 2a = 2 2 = 1
The y-coordinate of the vertex is
y = (1)2 -2(1) - 8 = -9.
The y-intercept is c = -8, and the x-intercepts are the solutions of
x2 - 2x - 8 = 0
(x + 2)(x - 4) = 0,
so that x = -2 and 4. Its graph is shown below. Top of Page
Exponential Functions

An exponential function is a function of the form

f(x) = Abx,
where A and b are constants and b > 0. (We call b the base of the exponential function.)
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"

 x y = 3(2x)

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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.

Law
Example
 bx by = bx+y
 23 22 = 25 = 32
 bx by = bx-y
 43 42 = 41 = 4
 1 bx = b-x
 1 90.5 = 9-0.5 = 1 3
 b0 = 1
 (3.3)0 = 1
 (bx)y = bx y
 (32)2 = 34 = 81
 (bc)x = bx cx
 (4 2)2 = 4222 = 64 b c x = bx cx 4 3 2 = 42 32 = 16 9

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Compound Interest

Future Value
If an amount P (the present value) earns interest at an annual interest rate r, compounded m times per year, then the accumulated amount (or future value) after t years is

 A = P 1 + r m m t
We often think of A as a function of t, and write
 A(t) = P 1 + r m m t
Example

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

P = 1000,     r = 0.048,     m = 12.
Substituting gives
A(t)=  1,000 1 + 0.048 12 12t
=1,000(1.004)12t.
This function gives the value of the investment after t years. For instance, after 5 years, the investment is worth
A(5) = 1000(1.004)12 5 = 1000(1.004) 60 = \$1270.64.

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The Number e

The numbers 1 + 1 m m
converge to e = 2.71828182845904523536. . . as m gets large. The following table shows the value of (1+1/m)m for several values of m. You can also add your own value of m and press "Compute" (note that very large values give computational errors --- experiment!).

m    1   10100100010000 1 + 1 m m
22.593742462.704813832.716923932.71814593

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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

A = Pert.
The effective yield from continuous compounding is given by
re = er - 1.

Example

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

A = 1000e0.048t.
The effective yield is
re = e0.048 - 1 ≈ 0.04917,
or 4.917% per year.

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Logarithms

The statement

logax = y
means that
ay = x.
log10x is usually written log x, and called the common logarithm. The expression logex is usually written as ln x and called the natural logarithm

Example

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

 Exponential Form 103 = 1000 42 = 16 51 = 5 70 = 1 4-2 = 1/16 Logarithm Form log 1000 = 3 log416 = 2 log55 = 1 log71 = 0 log4(1/16) =-2

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Logarithm Identities

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

Identity
Example
 loga(xy) = logax + logay
 log216 = log28 + log22
 loga(x/y) = logax - logay
 log2 (5/3) = log25 - log23
 loga(xr) = r logax
 log2(65) = 5 log26
 logaa = 1 loga1 = 0
 log22 = 1 log31 = 0
 loga(1/x) = -logax
 log2(1/3) = -log23
 logax = log x log a = ln x ln a
 log25 = log 5 log 2 ≈ 2.3219

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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

alogax= x
for all positive x and
loga(ax) = x
for all real x.

Want to learn more about inverse functions? Go to the on-line text on inverse functions.

Examples

2log2x = x
eln x = x
log2( 2x  ) = x
ln ( ex  ) = x

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Exponential Decay and Half-Life

An exponential decay function has the form

Q(t) = Q0e-kt           Q0, k both positive
Q0 represents the value of Q at time t = 0, and k is the decay constant.

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

th k = ln 2.

Exponential Growth and Doubling Time

An exponential growth function has the form

Q(t) = Q0ekt           Q0, k both positive
Q0 represents the value of Q at time t = 0, and k is the growth constant.

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

td k = ln 2.
Examples

Decay and Half-Life:

1. Q(t) = Q0e-0.000 120 968t is the decay function for carbon-14.

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:

1. P(t) = 1000e0.5t is the amount of money in an account after t years if \$1000 invested at 5% annually with interest compounded continuously.

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

 f(x) = N 1 + Ab-x (A, N, b constant, b positive and ≠ 1)

Properties of the Logistic Curve

• The graph is an S-shaped curve sandwiched between the horizontal lines y = 0 and y = N. N is called the limiting value of the logistic curve.
• If b > 1 the graph rises; if b < 1, the graph falls.
• The y intercept is N/(1 + A)
• Role of b: For small x, the logistic function grows approximately exponentially with base b, and follows the curve [N/(1+A)] bx. b > 1 0 < b < 1
Examples

N = 50, A = 24, b = 3 gives

 f(x) = 50 1 + 24(3-x)
Technology format: 50/(1+24*3^(-x))

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

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Last Updated: May 2007