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# Match the slope fields with their differential equations

By Zachary Parker,2014-11-25 17:57
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Match the slope fields with their differential equations

Match the slope fields with their differential equations.

(A) (B)

(C) (D)

dydydydy7. 8. 9. 10. xsinx，？2y，？xydxdxdxdx

____________________________________________________________________________ Match the slope fields with their differential equations.

(A) (B)

(C) (D)

dyxdydydy，？11. 12. 13. 14. ，？.51x.5y，？xydxydxdxdx

_____________________________________________________________________________ 15. (From the AP Calculus Course Description)

The slope field from a certain differential equation is shown above. Which of the following

could be a specific solution to that differential equation?

2xxyxyeyeyxln(A) (B) (C) (D) (E) yxcos

16.

The slope field for a certain differential equation is shown above. Which of the following could be a specific solution to that differential equation?

123yx(A) (B) (C) (D) (E) yxsinyxyxlnyxcos6

______________________________________________________________________________

dyxy17. Consider the differential equation given by . dx2

(a) On the axes provided, sketch a slope field for the given differential equation. (b) Let f be the function that satisfies the given differential equation. Write an equation for the

tangent line to the curve through the point (1, 1). Then use your tangent line yfx;；

equation to estimate the value of f1.2;；

(c) Find the particular solution to the differential equation with the initial yfx;；

condition . Use your solution to find . f11f1.2;；;；

(d) Compare your estimate of found in part (b) to the actual value of found in f1.2f1.2;；;；

part (c). Was your estimate from part (b) an underestimate or an overestimate? Use your

slope field to explain why.

______________________________________________________________________________

dyx18. Consider the differential equation given by . dxy

(a) On the axes provided, sketch a slope field for the given differential equation.

(b) Sketch a solution curve that passes through the point (0, 1) on your slope field. (c) Find the particular solution to the differential equation with the initial yfx;；

condition . f01;；

(d) Sketch a solution curve that passes through the point on your slope field. 0,1;；

(e) Find the particular solution to the differential equation with the initial yfx;；

condition . f01，？;；

dyx219. Consider the differential equation given by . 2dxx1

(a) On the axes provided, sketch a slope field for the given differential equation. (b) Sketch a solution curve that passes through the point (0, 1) on your slope field.

2dy(c) Find . For what values of x is the graph of the solution concave yfx;；2dx

up? Concave down?

____________________________________________________________________________

dy120. Consider the logistic differential equation ; ，？yy2;；dt2

(a) On the axes provided, sketch a slope field for the given differential equation.

(b) Sketch a solution curve that passes through the point (4, 1) on your slope field.

2y(c) Show that satisfies the given differential equation. te12

limy(d) Find by using the solution curve given in part (c). t，！

2dy(e) Find . For what values of y, 0< y < 2, does the graph of have an yft;；2dt

inflection point?

_____________________________________________________________________________

dP221. (a) On the slope field for ，？33PP, sketch three dt

solution curves showing different types of behavior

for the population P.

(b) Is there a stable value of the population? If so, what is it?

(c) Describe the meaning of the shape of the solution curves

for the population: Where is P increasing? Decreasing?

What happens in the long run? Are there any inflection

points? Where? What do they mean for the population?

dPdP (d) Sketch a graph of against P. Where is positive? dtdt

Negative? Zero? Maximum? How do your observations

dP about explain the shapes of your solution curves? dt

(Problem 21 is from Calculus (Third Edition) by Hughes-Hallett, Gleason, et al)

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