All Calculus 1 Resources
Example Question #1 : How To Find 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 .
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 .
1 is our final answer.
Example Question #2 : How To Find Prediction Models
Approximate the value at of the function ,with a linear approximation using the slope of the function at .
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
Example Question #3 : How To Find Prediction Models
Determine the tangent line to at , and use the tangent line to approximate the value at .
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, ,.
, where is the slope of the tangent line.
To use point-slope form, we need to know the value of the original function at ,