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School of Mathematics and Statistics
Carleton University
Math. 69.107

ASSIGNMENT 1
SOLUTIONS

  1. [5] Let tex2html_wrap_inline315 . Use the definition of the limit and the Binomial Theorem to calculate

    displaymath317

    Solution From the Binomial Theorem we know that tex2html_wrap_inline319 . Next, we have tex2html_wrap_inline321 or, since f(1) = 3-2+1 = 2, we have for tex2html_wrap_inline325 ,

    eqnarray16

    It follows that

    displaymath327

    Remark: As a check note that tex2html_wrap_inline329 by the Power Rule.

  2. [5]a)
    Let tex2html_wrap_inline331 Find the derivative of f, i.e., tex2html_wrap_inline335 , using any method whatsoever.
    [5]b)
    Evaluate tex2html_wrap_inline337 .

    Solution tex2html_wrap_inline339 where tex2html_wrap_inline341 . By the Generalized Power Rule, we get

    eqnarray48

    At x=0 we have an indeterminate form, 0/0, so we must rewrite the expression in order to evaluate the two-sided derivative as an ordinary limit. Note that

    displaymath347

    for x>0. Using this we see that the right-derivative of f at x=0 is given by

    eqnarray91

    The same argument can be used to show that

    displaymath355

    if x < 0 since tex2html_wrap_inline359 , in this case. So,

    eqnarray116

    Since the left and right-derivatives are not equal at x=0, it follows that tex2html_wrap_inline337 DOES NOT EXIST.

  3. [5]a)
    A function y = f(x) is defined implicitly by the relation

    displaymath367

    Find the equation of the tangent line to the graph of y = f(x) at the point tex2html_wrap_inline371 .

    [5]b)
    Calculate tex2html_wrap_inline373 at the point tex2html_wrap_inline371 .

    Solution y is given implicitly. So, the slope of the required tangent line is given by tex2html_wrap_inline379 where tex2html_wrap_inline381 is given by

    displaymath383

    Solving for tex2html_wrap_inline381 , we get

    displaymath387

    Thus, tex2html_wrap_inline389 . It follows that the tangent line looks like y-0 = 0(x-1) = 0. So y=0 (or the x-axis) is the equation of the tangent line.

    In order to evaluate tex2html_wrap_inline373 at the point tex2html_wrap_inline371 we know that

    displaymath401

    At x=1, y=0 we get that tex2html_wrap_inline405 .

  4. Find the derivatives of the following functions.
    [5] a)
    tex2html_wrap_inline407 ,
    [5] b)
    tex2html_wrap_inline409

    Solution a) tex2html_wrap_inline411 ,

    b) tex2html_wrap_inline413

  5. Evaluate the following limits (Show all work).
    [5] a)
    tex2html_wrap_inline415 ,
    [5] b)
    tex2html_wrap_inline417 .

    Solution a) tex2html_wrap_inline419 , since tex2html_wrap_inline421 as tex2html_wrap_inline423 . On the other hand, tex2html_wrap_inline425 as tex2html_wrap_inline423 . So their product must approach tex2html_wrap_inline429 .

    Alternately, tex2html_wrap_inline431

    b) We use the identity tex2html_wrap_inline433 with tex2html_wrap_inline435 . Then,

    eqnarray204

    But tex2html_wrap_inline437 as tex2html_wrap_inline439 . Furthermore tex2html_wrap_inline441 as tex2html_wrap_inline439 and so

    eqnarray219

    On the other hand,

    eqnarray231

    It follows that the required limit DOES NOT EXIST.

  6. [10] The problem of modeling planetary motion in the case of two bodies has been known since the time of Kepler and Newton. Using classical approximations it is known that planets will travel in ellipses with the sun at one focus. Asteroids or comets tend to travel in highly eccentric orbits (resembling parabolae) in the plane of the solar system with the sun at their focus.

    Let's assume that a planetary body is orbiting the sun (which is assumed to be very close to the origin) in a fixed almost circular orbit given by

    displaymath445

    Let's say that an asteroid is approaching the sun in an orbit whose equation is given by

    displaymath447

    Use Newton's method to find the two expected points of crossing of these two orbits, i.e., the two possible collision points, to three significant digits.

Solution Use tex2html_wrap_inline449 . Let tex2html_wrap_inline451 and tex2html_wrap_inline453 . The required roots are given by the zeros of h(x) = f(x) - g(x). Find tex2html_wrap_inline457 and then tex2html_wrap_inline459 . Then use the iteration

displaymath461

displaymath463

and the answer is 0.995 where there are two values for y given by tex2html_wrap_inline469 .

Total: /60



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Next: About this document

Angelo Mingarelli
Mon Oct 26 10:45:26 EST 1998