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Thread: Visual solution required for the problems

  1. #1

    Post Visual solution required for the problems

    1. If Modulus of Z is not equal to 1, then all the values of Z/1-Z2 lie on
    2. Consider f(x) = kex-x for all real x where k is a real constant.
    a. Where does line y=x meet kex for k <=0
    b. What is the positive value of k for which kex-x=0 has only one root.
    c. For k>0, find the roots of k for which kex-x=0

    3. Consider planes 3x-6y-2z=15, 2x+y-2z=5. Prove that vector 14i+2j+15k is parallel to the line of intersection of given planes
    4. Prove that the curve y=-x2/2 +x+1 is symmetric with respect to the line x=1
    5. Let f(x)= 2+cosx for all real x. Prove that for each real t, there exists a point c in (t, t+pi) such that f ‘ (c) = 0.

  2. #2
    Sketchpad Developer
    Join Date
    Oct 2010
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    Philadelphia
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    71
    Hi Gopalakrishnan,
    Sketchpad is not a program for doing proofs, but for constructing and exploring a wide range of mathematical objects. As such, it can shed some light on some of your questions, and enable a student to better visualize them.

    Unfortunately, I'm also uncertain of the details of some of your questions. For instance, if I interpret #1 to refer to complex variable Z, then a bit of experimentation with Sketchpad indicates that the range of f(Z) = Z/(1-Z^2) appears to include the entire complex plane. But I don't know if I've interpreted your question correctly.

    Similarly, #2 is easy to graph with Sketchpad, and it's easy to manipulate the value of k, and to find the intersection of the graph with the line y=x. This makes for an interesting exploration, and the observed behavior may help shed light on ways to obtain an analytic solution. But Sketchpad is not a computer algebra system, and cannot produce the analytic solution, even though the intersection of the two graphs provides a very accurate numeric solution for any value of k.

    With respect to #3, Sketchpad isn't a 3D visualization program, and though there are a number of 3D toolkits for Sketchpad that users have created, I don't know of any sketches designed to find the intersection of two planes.

    For #4, Sketchpad makes it easy to visualize the situation, to verify that the vertex of the parabola is at (1, 1.5), and to verify that the parabola is symmetric with respect to x=1. But Sketchpad does not prove this analytically; rather, it enables the investigation. This same comment applies to #5: Sketchpad is very useful for visualizing the problem, but it does not create a proof.

    I hope these comments are useful to you. If you like, I can share with you some of the visualizations that I created for questions 1, 2, and 4.
    --Scott

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