# Calculus 1 : Prediction Models

## Example Questions

### Example Question #1 : Prediction Models

Suppose you are a banker and set up a very unique function for your interest rate over time given by However, you find your computer incapable of calculating the interest rate at . Estimate the value of the interest rate at by using a linear approximation, using the slope of the function at .  Undefined  Explanation:

To do a linear approximation, we're going to create a function , that approximates our situation. In our case, m will be the slope of the function at , while b will be the value of the function at . The z will be distance from our starting position to our end position , which is Firstly, we need to find the derivative of with respect to x to determine slope.

By the power rule: The slope at will therefore be 0 since .

Since this is the case, the approximate value of our interest rate will be identical to the value of the original function at x=2, which is  ### Example Question #2 : Prediction Models

Approximate the value at of the function ,with a linear approximation using the slope of the function at      Explanation:

To do this, we must determine the slope of the function at , which we will call , and the initial value of the function at , which we will call , and since is only away from , our linear approximation will look like: To determine slope, we take the derivative of the function with respect to x and find its value at , which in our case is: At , our value for is To determine , we need to determine the value of the original equation at  At , our value for b is Since  ### Example Question #1 : How To Find Prediction Models

Determine the tangent line to at , and use the tangent line to approximate the value at .     Explanation:

First recall that To find the tangent line of at , we first determine the slope of . To do so, we must find its derivative.

Recall that derivatives of exponential functions involving are given as: , where is a constant and is any function of In our case, ,. At , , where is the slope of the tangent line.

To use point-slope form, we need to know the value of the original function at   Therefore, At   ### All Calculus 1 Resources 