Prediction Models - AP Calculus AB
Card 0 of 42
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 
.
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
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.
To do a linear approximation, we're going to create a function
Firstly, we need to find the derivative of
By the power rule:
The slope at
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.
Compare your answer with the correct one above
Approximate the value at 
of the function 
,with a linear approximation using the slope of the function at 
.
Approximate the value 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 
Since 
, 
To do this, we must determine the slope of the function at
To determine slope, we take the derivative of the function with respect to x and find its value at
At
To determine
At
Since
Compare your answer with the correct one above
Determine the tangent line to 
at 
, and use the tangent line to approximate the value at 
.
Determine the tangent line to
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 
,
First recall that
To find the tangent line of
Recall that derivatives of exponential functions involving
In our case,
At
To use point-slope form, we need to know the value of the original function at
Therefore,
At
Compare your answer with the correct one above
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 .
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
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.
To do a linear approximation, we're going to create a function
Firstly, we need to find the derivative of
By the power rule:
The slope at
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.
Compare your answer with the correct one above
Approximate the value at of the function ,with a linear approximation using the slope of the function at .
Approximate the value 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
Since ,
To do this, we must determine the slope of the function at
To determine slope, we take the derivative of the function with respect to x and find its value at
At
To determine
At
Since
Compare your answer with the correct one above
Determine the tangent line to at , and use the tangent line to approximate the value at .
Determine the tangent line to
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 ,
First recall that
To find the tangent line of
Recall that derivatives of exponential functions involving
In our case,
At
To use point-slope form, we need to know the value of the original function at
Therefore,
At
Compare your answer with the correct one above
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 .
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
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.
To do a linear approximation, we're going to create a function
Firstly, we need to find the derivative of
By the power rule:
The slope at
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.
Compare your answer with the correct one above
Approximate the value at of the function ,with a linear approximation using the slope of the function at .
Approximate the value 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
Since ,
To do this, we must determine the slope of the function at
To determine slope, we take the derivative of the function with respect to x and find its value at
At
To determine
At
Since
Compare your answer with the correct one above
Determine the tangent line to at , and use the tangent line to approximate the value at .
Determine the tangent line to
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 ,
First recall that
To find the tangent line of
Recall that derivatives of exponential functions involving
In our case,
At
To use point-slope form, we need to know the value of the original function at
Therefore,
At
Compare your answer with the correct one above
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 .
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
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.
To do a linear approximation, we're going to create a function
Firstly, we need to find the derivative of
By the power rule:
The slope at
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.
Compare your answer with the correct one above
Approximate the value at of the function ,with a linear approximation using the slope of the function at .
Approximate the value 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
Since ,
To do this, we must determine the slope of the function at
To determine slope, we take the derivative of the function with respect to x and find its value at
At
To determine
At
Since
Compare your answer with the correct one above
Determine the tangent line to at , and use the tangent line to approximate the value at .
Determine the tangent line to
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 ,
First recall that
To find the tangent line of
Recall that derivatives of exponential functions involving
In our case,
At
To use point-slope form, we need to know the value of the original function at
Therefore,
At
Compare your answer with the correct one above
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 .
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
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.
To do a linear approximation, we're going to create a function
Firstly, we need to find the derivative of
By the power rule:
The slope at
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.
Compare your answer with the correct one above
Approximate the value at of the function ,with a linear approximation using the slope of the function at .
Approximate the value 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
Since ,
To do this, we must determine the slope of the function at
To determine slope, we take the derivative of the function with respect to x and find its value at
At
To determine
At
Since
Compare your answer with the correct one above
Determine the tangent line to at , and use the tangent line to approximate the value at .
Determine the tangent line to
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 ,
First recall that
To find the tangent line of
Recall that derivatives of exponential functions involving
In our case,
At
To use point-slope form, we need to know the value of the original function at
Therefore,
At
Compare your answer with the correct one above
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 .
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
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.
To do a linear approximation, we're going to create a function
Firstly, we need to find the derivative of
By the power rule:
The slope at
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.
Compare your answer with the correct one above
Approximate the value at of the function ,with a linear approximation using the slope of the function at .
Approximate the value 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
Since ,
To do this, we must determine the slope of the function at
To determine slope, we take the derivative of the function with respect to x and find its value at
At
To determine
At
Since
Compare your answer with the correct one above
Determine the tangent line to at , and use the tangent line to approximate the value at .
Determine the tangent line to
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 ,
First recall that
To find the tangent line of
Recall that derivatives of exponential functions involving
In our case,
At
To use point-slope form, we need to know the value of the original function at
Therefore,
At
Compare your answer with the correct one above
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 .
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
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.
To do a linear approximation, we're going to create a function
Firstly, we need to find the derivative of
By the power rule:
The slope at
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.
Compare your answer with the correct one above
Approximate the value at of the function ,with a linear approximation using the slope of the function at .
Approximate the value 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
Since ,
To do this, we must determine the slope of the function at
To determine slope, we take the derivative of the function with respect to x and find its value at
At
To determine
At
Since
Compare your answer with the correct one above