All Calculus AB Resources
Example Questions
Example Question #1 : Model And Verify Differential Equations
What is a differential equation?
A function that has both positive and negative intervals in its domain
An equation that is completely negative
An equation with a function and a least one of its derivatives
An equation in which we are taking the difference of two functions
An equation with a function and a least one of its derivatives
Let’s say we have a function and that function plus its derivative is a solution to our function . This gives us the differential equation:
This is a differential equation because it has both the function and its derivative as part of the equation.
Example Question #1 : Model And Verify Differential Equations
Which of the following is a differential equation?
Recall that a differential equation is an equation that has both the function and at least one of its derivatives in the equation. The last answer, , is the only equation that has both the function and the derivative in the equation so this is our answer.
Example Question #3 : Model And Verify Differential Equations
Is the equation a solution to the differential equation ?
Yes
No
There is not enough information
No
First we need to take the derivative with respect to of our function .
Now we can plug this back into our differential equation that we were given.
To simplify things further, let’s also write in terms of . So our original function says . We will plug in for all terms in the differential equation.
And so is not a solution to the differential equation .
Example Question #4 : Model And Verify Differential Equations
Is the equation a solution to the differential equation ?
No
There is not enough information
Yes
Yes
We begin by taking the derivative with respect to of our function .
Next we will plug this value into our differential equation.
Next we will write all in terms of . Recall that our original function says that , we will plug this in for all terms.
And so , is a solution to the differential equation .
Example Question #3 : Differential Equations
Which of the following is a solution for the differential equation ?
Let us consider . By taking the derivative we see that . We will plug this back into the differential equation as well as subbing in .
And so is a solution to the differential equation
Example Question #1 : Model And Verify Differential Equations
True or False: All differential equations will have a solution to them.
True
False
False
Not all differential equations will have solutions. Many, if not all, first order differential equations will have solutions to them. As we move into second and third order differential equations, however, some of these may have no solutions.
Example Question #2 : Model And Verify Differential Equations
True or False: is a differential equation.
False
True
True
This is a second order differential equation. Recall that differential equations are equations that have both a function and AT LEAST ONE of their derivatives. Equations with more than one derivative, such as those with first, second, and third order derivatives, are still differential equations. This second order differential equation includes the second order derivative.
Example Question #561 : Calculus Ab
Which of the following is a solution to the differential equation ?
Let’s consider the equation . If we take the derivative we see that . Let’s plug that in for in our differential equation as well as substitute .
And so the function is a solution to the differential equation .
Example Question #6 : Model And Verify Differential Equations
Which of the following is a solution to the differential equation
Example Question #1 : Model And Verify Differential Equations
Which of the following is an example of when we would use a differential equation in real life?
None of these
To find the slope of a mountain
To predict or simulate a population’s growth
To find the height of a flag pole using trigonometric information
To predict or simulate a population’s growth
Of the above examples, this is the only one that has a rate of change over time. A population’s growth will vary over time depending on several factors such as resources available, predator/prey interactions, and carrying capacity. The exponential growth model is in fact a differential equation:
Where is the growth rate and is the current population. This differential equation has the solution . Often times (if not all) population’s cannot grow exponentially forever, and so we also have a differential equation that is the logistic growth model which takes into account carrying capacity:
Where is the carrying capacity. The solution to this differential equation is .
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