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Four Problems on Linear Approximation

  1. Jul 13, 2008 #1
    On my last test I got four problems wrong. I'd like to know what I did wrong on these for my final.

    1. Given f(x) = x^(3/2) ; x=4; and delta x = dx = 0.1; calculate delta y
    2. Use differentials to approximate the change in f(x) if x changes from 3 to 3.01 and f(x) = (3x^2-26)^10
    3. f(x) = x^(-1/3); approximate (7.952)^(-1/3)
    4. The equatorial radius of the earth is approx 3690 miles. Suppose a wire is wrapped tightly around the earth at its equator. How much must this wire be lengthened if it is to be strung on poles 10 feet above the ground.

    My solutions
    delta y = f(x+ delta x) - f(x) = 4.1^(3/2) - 0.1^(3/2)

    f'(x)dx = 10(1)^9 times 0.01 = 0.1

    f(8) - f'(8)(0.048) = 1/2 - (-1/3)(8^(-4/3))(0.048) = 1/2 - (1/48)(0.048) = 1/2 - 0.001 = 499/1000

    f(x)=2 pi r
    f'(x) = 2 pi
    f'(x)dx = 2 pi times 10 = 20 pi

    If you could help me with even one of the problems, I'll be happy. I think I might be making the same time of mistake, since the problems are so similar.
  2. jcsd
  3. Jul 13, 2008 #2
    problem 1: you want f(x+dx)-f(x)... You got f(x+dx) right, but look close at what you got for f(x)
    problem 3: you lost a negative sign right at the end.
    problem 4: be careful with your notation! these are not functions of x, but r. Also, in the problem, you're given that r is 3690 miles, but dr is 10 feet, you need units in your answer.

    These are all pretty subtle mistakes and don't betray any lack of understanding of the technique. Everyone does it, so don't worry.

    Problem 2, however is a bit more serious... You need to remember to use the chain rule to differentiate, f'(x) = 10*(3x^2-26)^9 * d/dx(3x^2-26)
  4. Jul 13, 2008 #3
    Thanks for your help. I've got this worked out now.
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