# 4.7 Exponential and logarithmic models  (Page 8/16)

 Page 8 / 16

## Verbal

With what kind of exponential model would half-life be associated? What role does half-life play in these models?

Half-life is a measure of decay and is thus associated with exponential decay models. The half-life of a substance or quantity is the amount of time it takes for half of the initial amount of that substance or quantity to decay.

What is carbon dating? Why does it work? Give an example in which carbon dating would be useful.

With what kind of exponential model would doubling time be associated? What role does doubling time play in these models?

Doubling time is a measure of growth and is thus associated with exponential growth models. The doubling time of a substance or quantity is the amount of time it takes for the initial amount of that substance or quantity to double in size.

Define Newton’s Law of Cooling. Then name at least three real-world situations where Newton’s Law of Cooling would be applied.

What is an order of magnitude? Why are orders of magnitude useful? Give an example to explain.

An order of magnitude is the nearest power of ten by which a quantity exponentially grows. It is also an approximate position on a logarithmic scale; Sample response: Orders of magnitude are useful when making comparisons between numbers that differ by a great amount. For example, the mass of Saturn is 95 times greater than the mass of Earth. This is the same as saying that the mass of Saturn is about $\text{\hspace{0.17em}}{10}^{\text{2}}\text{\hspace{0.17em}}$ times, or 2 orders of magnitude greater, than the mass of Earth.

## Numeric

The temperature of an object in degrees Fahrenheit after t minutes is represented by the equation $\text{\hspace{0.17em}}T\left(t\right)=68{e}^{-0.0174t}+72.\text{\hspace{0.17em}}$ To the nearest degree, what is the temperature of the object after one and a half hours?

For the following exercises, use the logistic growth model $\text{\hspace{0.17em}}f\left(x\right)=\frac{150}{1+8{e}^{-2x}}.$

Find and interpret $\text{\hspace{0.17em}}f\left(0\right).\text{\hspace{0.17em}}$ Round to the nearest tenth.

$f\left(0\right)\approx 16.7;\text{\hspace{0.17em}}$ The amount initially present is about 16.7 units.

Find and interpret $\text{\hspace{0.17em}}f\left(4\right).\text{\hspace{0.17em}}$ Round to the nearest tenth.

Find the carrying capacity.

150

Graph the model.

Determine whether the data from the table could best be represented as a function that is linear, exponential, or logarithmic. Then write a formula for a model that represents the data.

 $x$ $f\left(x\right)$ –2 0.694 –1 0.833 0 1 1 1.2 2 1.44 3 1.728 4 2.074 5 2.488

exponential; $\text{\hspace{0.17em}}f\left(x\right)={1.2}^{x}$

Rewrite $\text{\hspace{0.17em}}f\left(x\right)=1.68{\left(0.65\right)}^{x}\text{\hspace{0.17em}}$ as an exponential equation with base $\text{\hspace{0.17em}}e\text{\hspace{0.17em}}$ to five significant digits.

## Technology

For the following exercises, enter the data from each table into a graphing calculator and graph the resulting scatter plots. Determine whether the data from the table could represent a function that is linear, exponential, or logarithmic.

 $x$ $f\left(x\right)$ 1 2 2 4.079 3 5.296 4 6.159 5 6.828 6 7.375 7 7.838 8 8.238 9 8.592 10 8.908

logarithmic

 $x$ $f\left(x\right)$ 1 2.4 2 2.88 3 3.456 4 4.147 5 4.977 6 5.972 7 7.166 8 8.6 9 10.32 10 12.383
 $x$ $f\left(x\right)$ 4 9.429 5 9.972 6 10.415 7 10.79 8 11.115 9 11.401 10 11.657 11 11.889 12 12.101 13 12.295

logarithmic

 $x$ $f\left(x\right)$ 1.25 5.75 2.25 8.75 3.56 12.68 4.2 14.6 5.65 18.95 6.75 22.25 7.25 23.75 8.6 27.8 9.25 29.75 10.5 33.5

For the following exercises, use a graphing calculator and this scenario: the population of a fish farm in $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ years is modeled by the equation $\text{\hspace{0.17em}}P\left(t\right)=\frac{1000}{1+9{e}^{-0.6t}}.$

difference between calculus and pre calculus?
give me an example of a problem so that I can practice answering
x³+y³+z³=42
Robert
dont forget the cube in each variable ;)
Robert
of she solves that, well ... then she has a lot of computational force under her command ....
Walter
what is a function?
I want to learn about the law of exponent
explain this
what is functions?
A mathematical relation such that every input has only one out.
Spiro
yes..it is a relationo of orders pairs of sets one or more input that leads to a exactly one output.
Mubita
Is a rule that assigns to each element X in a set A exactly one element, called F(x), in a set B.
RichieRich
If the plane intersects the cone (either above or below) horizontally, what figure will be created?
can you not take the square root of a negative number
No because a negative times a negative is a positive. No matter what you do you can never multiply the same number by itself and end with a negative
lurverkitten
Actually you can. you get what's called an Imaginary number denoted by i which is represented on the complex plane. The reply above would be correct if we were still confined to the "real" number line.
Liam
Suppose P= {-3,1,3} Q={-3,-2-1} and R= {-2,2,3}.what is the intersection
can I get some pretty basic questions
In what way does set notation relate to function notation
Ama
is precalculus needed to take caculus
It depends on what you already know. Just test yourself with some precalculus questions. If you find them easy, you're good to go.
Spiro
the solution doesn't seem right for this problem
what is the domain of f(x)=x-4/x^2-2x-15 then
x is different from -5&3
Seid
All real x except 5 and - 3
Spiro
***youtu.be/ESxOXfh2Poc
Loree
how to prroved cos⁴x-sin⁴x= cos²x-sin²x are equal
Don't think that you can.
Elliott
By using some imaginary no.
Tanmay
how do you provided cos⁴x-sin⁴x = cos²x-sin²x are equal
What are the question marks for?
Elliott