Exponential functions (calculator exercise)

In summary, the conversation discusses solving an equation involving exponential functions and finding the maximum point on a graph of another function. The equation cannot be solved algebraically and requires the use of a CAS for numeric approximations. The graph shows the maximum point at (50.1865, 27.1906).
  • #1
Joshuaniktas
1
0
Hi there, I have tried to do these questions but I don't understand. Any help would be appreciated!

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  • #2
Hello, and welcome to MHB! :)

Let's begin with part (a). We are to find when:

\(\displaystyle f(t)=g(t)\)

Or:

\(\displaystyle e^{-\frac{t}{20}}+10=te^{-\frac{t}{20}}+6\)

Let's subtract 6 from both sides:

\(\displaystyle e^{-\frac{t}{20}}+4=te^{-\frac{t}{20}}\)

And then arrange as:

\(\displaystyle 4=te^{-\frac{t}{20}}-e^{-\frac{t}{20}}\)

[DESMOS]{"version":7,"graph":{"viewport":{"xmin":-34.31311077614891,"ymin":-11.279300231558665,"xmax":104.86022436822739,"ymax":62.76481974796097}},"randomSeed":"a115e9332f088699d9a4c7866dadfcce","expressions":{"list":[{"type":"expression","id":"1","color":"#c74440","latex":"y=4e^{\\frac{t}{20}}"},{"type":"expression","id":"2","color":"#2d70b3","latex":"y=t-1"}]}}[/DESMOS]

As we cannot solve this algebraically, we will need to rely on a CAS to generate numeric approximations:

https://www.wolframalpha.com/input/?i=4e^(t/20)=t-1
We get:

\(\displaystyle t\approx6.54997\)

\(\displaystyle t\approx50.1865\)

Now, for part (b), let's look at this graph:

[DESMOS]{"version":7,"graph":{"viewport":{"xmin":-9.635503167985057,"ymin":-15.7165503991205,"xmax":71.01286536037324,"ymax":27.190644928716516}},"randomSeed":"5093a3ded45d142de4030d436ace2162","expressions":{"list":[{"type":"expression","id":"1","color":"#c74440","latex":"y=xe^{-\\frac{x}{20}}+6\\left\\{0\\le x\\le60\\right\\}"}]}}[/DESMOS]

Click on the function's definition on the left to make it active, and you will see the maximum point on which you can click...what do you see when the point is labeled?
 

FAQ: Exponential functions (calculator exercise)

What is an exponential function?

An exponential function is a mathematical function of the form f(x) = ab^x, where a and b are constants and x is the independent variable. This type of function is characterized by a constant ratio between the output and the input, resulting in a curve that increases or decreases rapidly.

How do you graph an exponential function on a calculator?

To graph an exponential function on a calculator, you first need to enter the function into the calculator using the appropriate syntax. Then, you can use the graphing function of the calculator to plot the points and draw the curve. Some calculators also have a built-in table function that can display a table of values for the function.

What is the difference between an exponential function and a linear function?

The main difference between an exponential function and a linear function is that the rate of change, or slope, in an exponential function is not constant. In a linear function, the rate of change is constant, resulting in a straight line. Additionally, exponential functions have a constant ratio between the output and the input, while linear functions have a constant difference between the output and the input.

How do you solve exponential functions using a calculator?

To solve exponential functions using a calculator, you can use the exponential function button (usually denoted as "exp" or "e^x") to input the function into the calculator. Then, you can use the calculator's solve function to find the value of the independent variable that satisfies the equation. Some calculators also have a logarithm function that can be used to solve exponential functions.

What are some real-life applications of exponential functions?

Exponential functions are commonly used in fields such as finance, biology, and physics. Some real-life applications include population growth, compound interest, radioactive decay, and bacterial growth. They can also be used to model the spread of diseases, the growth of technology, and the decay of natural resources.

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