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and Exponential

Functions

TeacherExcellenceWorkshop

TeacherExcellenceWorkshop

D. P. Dwiggins, PhD

Department of

Mathematical Sciences

July 20, 2008

July 20, 2008

- Properties of Exponential Functions
- Properties of Logarithmic Functions
- As Inverses of Exponential Functions
- Logarithmic and Exponential Graphs
- Logarithm as Area
- Slopes of Tangent Lines for Logarithmic and Exponential Curves
- Definition of the Natural Exponential Base
- Applications
- Population Growth Models
- Radioactive Decay
- Newton’s Law of Cooling

In this list of rules, a denotes a positive real number, called the base of the exponential function.

We could use a particular base (such as 2 or 10), and we can choose any letter to stand for the base.

In the next slide, the base will be denoted by the letter e.

Exponential functions turn sums intoproducts, so that additive properties become multiplicative properties.

The additive identity (zero) becomesthe multiplicative identity (one).

The additive inverse (negative) becomesthe multiplicative inverse (reciprocal).

Inverse addition (subtraction) becomesinverse multiplication (division).

Multiplication becomes exponentiation.

Let E(x) denote any function which turns sums into products:

Setting y = 0 into the above equation gives

which means

must be true for every x.But the only way for this to be true is if

Turn Sums Into Products

Therefore exponential functions turn the additive identityinto the multiplicative identity.

Let E(x) denote any function which turns sums into products:

Setting y = –x into the above equation gives

which means

But E(0) = 1, and so E(x) and E(–x) must bemultiplicative reciprocals. That is,

Turn Sums Into Products

Exponential functions turn additive inverses into multiplicative inverses.

Let E(x) denote any function which turns sums into products:

Setting y = x into the above equation gives

which means

Turn Sums Into Products

Exponentiation increases the order of arithmetic,turning simple operations into more complicated ones.Exponential functions turn addition into multiplication,and so they also turn multiplication into exponentiation.

of Exponential Functions

Every exponential function is one-to-one:

Thus, every exponential function E(x) has an inverse,and the inverse of an exponential function definesa logarithmic function L(x),

In terms of a specific base (a) for the exponentialfunction, this statement becomes

Let L(x) denote any function which turns products into sums:

Setting y = 1 into the above equation gives

which means

must be true for every x.But the only way for this to be true is if

Turn Products Into Sums

Therefore logarithmic functions turn the multiplicativeidentity into the additive identity.

Let L(x) denote any function which turns products into sums:

Setting y = 0 into the above equation gives

which means

must be true for every x.

Turn Products Into Sums

But the only way for this to be true (if L(0) is defined)is if L(x) = 0 for every x. Thus, in order to make thelogarithmic function nontrivial, we have to assume

Let L(x) denote any function which turns products into sums:

Setting y = 1/x into the above equation gives

Turn Products Into Sums

which means

But L(1) = 0, and so L(x) and L(1/x) must benegatives of each other. That is,

Exponential functions turn multiplicative inverses into additive inverses.

Let L(x) denote any function which turns products into sums:

Setting y = x into the above equation gives

which means

Turn Products Into Sums

Taking logarithms decreases the order of arithmetic,turning complicated operations into simpler ones.Logarithmic functions turn multiplication into addition, andso they also take away exponents and turn them into products.

Exponential Function

Let E(x) be any exponential function. To calculate itsderivative, we begin by calculating the difference quotient:

In other words, the derivative of any exponential functionis equal to itself, multiplied by a constant scale factor,and this constant is the slope of the tangent line at (0,1).

Exponential Function

The exponential function which has the property that theslope of the tangent line at (0,1) has the value m0 = 1is called the natural exponential function, written asexp(x) or more frequently as ex, where e is the base ofthe exponential function which has unit slope at (0,1).

ex has the property of being equal to its own derivative:

This means y = ex solves the differential equation y = y.In general any exponential function solves a diff. eq.of the form y = ky, where k again is the slope of thetangent line at (0,1). Using the chain rule from calculus,we can show any such function is of the form y = ekx.

Logarithmic Function

Let y = L(x), corresponding to x = E(y).Since the derivative of E is equal to itself times m0,

as the derivative of the logarithmic function L(x).

In particular, when m0 = 1, so that L(x) becomes thenatural logarithmic function ln(x), we have

Since L(x) is defined for x > 0, the above shows L always hasa positive derivative, and so logarithms are strictly increasing.

Logarithmic Function

we can use logarithms to calculate definite integrals wherethe integrand is a reciprocal function. For example,

Many textbooks turn this equation around, switching x and t,and write a “definition” of the natural logarithm as

Textbooks then start with this and derive all the properties oflogarithmic functions, which is completely backwards, sincewe have just shown that this “definition” follows from thealgebraic properties of exponential and logarithmic functions.

Population Growth

Suppose P(t) represents the size of a growing population P,with initial size P = P0 at t = 0. The basic assumption of apopulation growth model is that the rate at which P changesis proportional to the size of P. In terms of differentials,

This last integral is just kt,

while the first integral is logarithmic. Including a constant ofintegration gives lnP + C = kt, where the constant C is evaluatedusing the initial condition: P = P0 at t = 0 C = – lnP0. Thus,

Radioactive Decay

The population growth model P = P0ektrepresents a quantitywhich grows exponentially with time for any positive k > 0.If k < 0, this model represents a quantity which gets smallerexponentially in time. In this case, k (or –k) is called thedecay constant, and this model represents radioactive decay.

Usually in this type of application the decay constant k isnot given; instead, what is given is the time t that it takes forhalf of the initial quantity to decay. (t is called the half-life.)

Newton’s Law of Cooling

Another application of the decay model is to consider thequantity Q = Q – QR, where Q is the temperature of a hotobject which is cooling down to room temperature, QR.

where Q0 is the object’s original temperature at time t = 0.

Problems utilizing all three types of these models are given on the next slide.

Problems

Suppose an initial population of 5,000 moves into an area, and it has a growth rate which causes it to double every ten years. What will the population be 25 years after it initially moved in?When will the population reach 25,000?

# 1.

The radioactive isotope found in corbomite has a half-life of 5,000 years. An artifact containing corbomite is discovered, and after measurement it is found to contain only 10% of the isotope as would be expected upon comparison with a modern sample of corbomite. How old is the artifact?

# 2.

A freshly made cup of coffee has an initial temperature of 90ºC. After one minute, the temperature has cooled down to 60ºC. If the room temperature is 20ºC, how long will it take for the coffee to cool down to body temperature? (Body temperature = 37ºC.)

# 3.

Given a collection of data,

Impose a linear relation on the data points:

Find an equation for the line by “regressing” the data points back to the imposed linear relation:

denote the error

between the predicted value for Yi and the

actual value (ei is also called the residual).

Let S denote a measure of the total error between

the data points and the regression line.

will not work, as the positive errors will cancel the negative errors and this sum will always equal zero.

is a possibility, but awkward to use because the absolute value function is non-differentiable at its local minimum.

Least Squares Method

Least Squares Regression finds the equation

for the regression line by minimizing the

sum of the squares of the errors:

Least Squares Method

S is minimized by taking derivatives of S with respect to A and B, setting these derivatives equal to zero, and solving the resulting equations for A and B.

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