Math Problem Statement
Find the exponential function y=y0e^(kt), with points (3,5) and (0,2)
Solution
To find the exponential function of the form using the points and , follow these steps:
Step 1: Use the point
From the equation , when , the equation becomes:
Thus, .
Step 2: Use the point
Now, substitute the values and into the equation , where :
Solve for :
Step 3: Solve for
Take the natural logarithm of both sides to solve for :
Using a calculator, we get:
Final Answer:
The exponential function is:
If you'd like further details or have any questions, feel free to ask!
Here are some related questions you may find interesting:
- How does changing the value of affect the graph of the exponential function?
- What is the behavior of the function as ?
- How can we derive exponential growth or decay from the general equation?
- What happens if is negative in this exponential model?
- How would the function change if the point (0, 2) was replaced by (0, 5)?
Tip: To understand exponential functions better, plot them for different values of to see how the growth rate changes.
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Math Problem Analysis
Mathematical Concepts
Exponential Functions
Logarithms
Algebra
Formulas
y = y0 e^(kt)
ln(x)
Theorems
Properties of Exponential Functions
Natural Logarithm Properties
Suitable Grade Level
Grades 10-12