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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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