# 9.2.2 AP Calculus Exam Slope Fields

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• karush
In summary, the slope field is a way to solve equations using a graph. The tangent line to y=f(x) at (a, b) is y=f'(a)(x- a)+ b. In (b) you are told that a=0 and b=1. What is f'(0) when you are also told that dy/dx= (3- y)cos(x)? Using that equation, what is y when x= 1?
karush
Gold Member
MHB
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I'm just going to post this image now since my tablet won't render the latex. This is a free response question..
But my experience is that the methods of solving are more focused here at mhb saving many error prone steps..

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What is your purpose in posting this? Do you just want someone to do your homework for you?

If you understand what a "slope field" is then (a) should be straight forward.

The tangent line to y= f(x) at (a, b) is y= f'(a)(x- a)+ b. In (b) you are told that a= 0 and b= 1. What is f'(0) when you are also told that dy/dx= (3- y)cos(x)? Using that equation, what is y when x= 1?

The equation, $$\frac{dy}{dx}= (3- y)cos(x)$$ is "separable" as $$\frac{dy}{3- y}= cos(x) dx$$. Integrate!

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It's not a homework assignment

View attachment 9323
(b) $f(0)=1$

$\dfrac{dy}{dx}\bigg|_{(0,1)} = (3-1)\cos(0) = 2$

tangent line at $(0,1)$ is $y-1 = 2(x-0) \implies y = 2x+1$

$f(0.2) \approx y = 2(0.2)+1 = 1.4$

(c) $\dfrac{dy}{3-y} = \cos{x} \, dx$

$\dfrac{dy}{y-3} = -\cos{x} \, dx$

$\ln|y-3| = -\sin{x} + C$

$y-3 = e^{-\sin{x} + C} = e^C \cdot e^{-\sin{x}} = Ae^{-\sin{x}}$

$y = 3+Ae^{-\sin{x}}$

initial condition is $(0,1)$ ...

$1 = 3 + Ae^0 \implies A = -2$

$f(x) = 3-2e^{-\sin{x}}$

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skeeter said:
(b) $f(0)=1$

$\dfrac{dy}{dx}\bigg|_{(0,1)} = (3-1)\cos(0) = 2$

tangent line at $(0,1)$ is $y-1 = 2(x-0) \implies y = 2x+1$

$f(0.2) \approx y = 2(0.2)+1 = 1.4$

(c) $\dfrac{dy}{3-y} = \cos{x} \, dx$

$\dfrac{dy}{y-3} = -\cos{x} \, dx$

$\ln|y-3| = -\sin{x} + C$

$y-3 = e^{-\sin{x} + C} = e^C \cdot e^{-\sin{x}} = Ae^{-\sin{x}}$

$y = 3+Ae^{-\sin{x}}$

initial condition is $(0,1)$ ...

$1 = 3 + Ae^0 \implies A = -2$

$f(x) = 3-2e^{-\sin{x}}$
Ok I really appreciate the help
I've always had difficulty in understanding slope Fields
I'll do some more and see if I can go thru it all the way.

I am basically reviewing this it never showed up when I took the class

## 1. What is a slope field?

A slope field is a graphical representation of the slopes of a differential equation at various points on a coordinate plane. It is used to visualize the behavior of a function and can help in finding solutions to differential equations.

## 2. How are slope fields used in the AP Calculus exam?

Slope fields are used in the AP Calculus exam to test students' understanding of differential equations and their ability to interpret and analyze graphical representations of functions. They may be asked to sketch a slope field, identify solutions to a given differential equation, or use a slope field to find the behavior of a function.

## 3. Can a slope field be used to find the exact solution to a differential equation?

No, a slope field cannot be used to find the exact solution to a differential equation. It can only provide an approximation of the behavior of a function. To find the exact solution, additional techniques such as separation of variables or Euler's method may be needed.

## 4. How is the direction of a slope field determined?

The direction of a slope field is determined by the slope of the tangent line at each point on the coordinate plane. This slope is calculated using the given differential equation and the coordinates of the point.

## 5. Are there any common patterns or behaviors that can be observed in slope fields?

Yes, there are certain common patterns and behaviors that can be observed in slope fields. These include horizontal and vertical asymptotes, equilibrium solutions, and periodic behavior. Identifying these patterns can help in understanding the behavior of a function and finding its solutions.

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