9.2.2 AP Calculus Exam -- slope field for which DE

In summary, the conversation discusses different methods for solving a problem, with one option being through observation and the other through separating variables and using integration. The end goal is to find the equation for a line, which is determined to be y = -x-1 or y+x=-1. The conversation also mentions the importance of noting the slopes near the line and how they are close to a value of -1.
  • #1
karush
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ok probably if one did a lot of these this could be solved by observation
others separate the variables and take to Integral to get the equation
 

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  • #2
karush said:
ok probably if one did a lot of these this could be solved by observation
others separate the variables and take to Integral to get the equation

meant to be done by observation ... note the slopes near this line are close to a value of -1

the equation of the line is about $\color{red}y = -x-1 \implies y+x=-1$

what do you think?
 

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  • #3
that looks like C

but how would you it was a line?
 

FAQ: 9.2.2 AP Calculus Exam -- slope field for which DE

1. What is a "slope field" in the context of the AP Calculus Exam?

A slope field is a graphical representation of the slope of a differential equation (DE) at various points in the xy-plane. It is used to visualize the behavior of a DE and can help in solving and understanding the solution to the DE.

2. How is a slope field created for a given DE?

To create a slope field, you first need to find the derivative of the DE. Then, for each point in the xy-plane, you plot a small line segment with a slope equal to the value of the derivative at that point. This process is repeated for multiple points to create a visual representation of the slope of the DE at different points.

3. Why is a slope field useful in solving DEs?

A slope field provides a visual representation of the behavior of a DE, which can help in understanding the solution to the DE. It can also be used to estimate the shape of the solution curve and identify any critical points or regions of interest.

4. Can a slope field be used to find the exact solution to a DE?

No, a slope field cannot be used to find the exact solution to a DE. It only provides an approximate visual representation of the behavior of the DE. To find the exact solution, other methods such as separation of variables or using an integrating factor may be necessary.

5. How can a slope field be used to check the accuracy of a solution to a DE?

A slope field can be used to check the accuracy of a solution to a DE by comparing the slope field to the solution curve. If the solution curve matches the slope field, it is likely that the solution is correct. If there are discrepancies, it may indicate an error in the solution or the need for further analysis.

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