### All Calculus 1 Resources

## 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 .

**Possible Answers:**

Undefined

**Correct answer:**

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 #1 : Prediction Models

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

**Possible Answers:**

**Correct answer:**

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 .

**Possible Answers:**

**Correct answer:**

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 ,