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  1. E

    Removable discontinuity solution

    For a function to be continuous at a point c, three conditions must be met: 1) f(c) is defined. The left graph in my attachment shows an example of a graph where f(c) is NOT defined. 2) lim_{x \rightarrow c} f(x) exists. The middle graph shows an example of a graph where f(c) is defined, but...
  2. E

    What rule have they used to change the integral?

    It's called "adding zero." :tongue: I see it more as a trick to eventually introduce a factor that can be canceled. For instance, if the integral was this: \int \frac{3x}{(x-2)^2} dx I would subtract and add 6: = \int \frac{3x - 6 + 6}{(x-2)^2} dx = \int \frac{3x - 6}{(x-2)^2} dx + \int...
  3. E

    Integral error

    If dv = e^{ax} dx, then v = e^{ax} is wrong. There are also multiple errors on the second line, but you need to fix what I said first.
  4. E

    Derivatives with constants

    Perhaps, but I have seen students who do, however, use a quotient rule even if the denominator is a constant. Weird, for sure... (shrugs)
  5. E

    Proof by induction sequence

    Best I can do is give you some other examples that use the properties that you may need to continue: 4(2+5^{n-3}) = 8 + 4 \cdot 5^{n-3} -7 \cdot 4^{n-1} = -7 \cdot 4 \cdot 4^{n-2} = -28 \cdot 4^{n-2} 12 \cdot 10^{n-5} - 4 \cdot 10^{n-5} = 8 \cdot 10^{n-5} Study the above and try your...
  6. E

    Proof by induction sequence

    5 \cdot 2^{n-2} \ne 10^{n-2} 5 \cdot 3^{n-2} \ne 15^{n-2} ...and so on. I don't know if this will help, but note that 2^{n-2} = 2\cdot 2^{n-3} and 3^{n-2} = 3\cdot 3^{n-3}
  7. E

    Function and relations

    Isn't the notation wrong? It looks like you want (f \circ g)(x), (g \circ f)(x) and (f \circ h)(x) (function composition) but it looks more like (fg)(x), (gf)(x) and (fh)(x) (combining functions by multiplication)
  8. E

    Algebra/Trigonometey (Precal) books with rigor?

    How about another old one: Principles of Mathematics by Allendoerfer/Oakey?
  9. E

    Linearization of a function

    I was told that the linearization is defined this way: L(x) = f(a) + f'(a)(x - a), where f is differentiable at a.
  10. E

    Local Extrema

    I knew people who were fooled by this sort of problem. It is possible that a local maximum be "lower" than a local minimum. Look at the graphs of secant or cosecant, for example.
  11. E

    Finding the volume using cylindrical shells

    Almost. Switch the order of subtraction.
  12. E

    Finding the volume using cylindrical shells

    No, you don't want to add 2 and 1 + (y - 2)2.
  13. E

    Finding the volume using cylindrical shells

    (To help visualize what you did earlier, I attached two pics. The red region is what is being rotated around the x-axis. The "Wrong.bmp" file shows what you did, and the "Right.bmp" file shows what the problem is asking.) The way you had set it up, your heights of the representative rectangle...
  14. E

    Finding the volume using cylindrical shells

    Here's where you've gone wrong, I think. The representative rectangles need to be inside the parabola, not outside, as you have set the integral up.
  15. E

    Emptying a Tank of Beer

    Whew! Glad I'm not totally out of it. Thanks.
  16. E

    Emptying a Tank of Beer

    So a cousin has asked me for Calculus help, but my Calculus is rusty. She's in Calculus II (of a 3-semester sequence in the US) and is on Work. I decided to make up a problem for her, but I want to make sure I know what I'm doing. 1. Homework Statement A cylindrical tank (16 feet high with a...
  17. E

    Finding Values that Satisfy a Limit

    That's what I thought -- I'm just too d***ed sleep-deprived. Thanks for the confirmation.
  18. E

    Finding Values that Satisfy a Limit

    This may be a dumb question, but I'll ask anyway... 1. Homework Statement Find the values of a and b such that \lim_{x \rightarrow 0} \frac{\sqrt{a + bx} - \sqrt{3}}{x} = \sqrt{3} 2. Homework Equations N/A 3. The Attempt at a Solution I already have the work and the solution...
  19. E

    Pre Calculus online with 2nd semester Algebra 2

    Yikes. In some schools all of that should be covered in Algebra 2. In my case, I remember learning conics, logarithms, and exponential functions in Algebra 2. We didn't get to matrices, though (and I didn't see matrices until senior year!). In the eyes of some schools, however, you did take...
  20. E

    Solve Inequalities algebraically

    I wouldn't solve it "algebraically." I would solve it by making a two-column table. First, find values of r that would make the left side 0 or undefined (I call these critical values), which appears you have already done. Then in the first column list all critical values and all intervals...
  21. E

    Radical equation

    2x^2 - 5x - 7 does not factor into (x - 7)(x + 2)! (x - 7)(x + 2) = x^2 - 5x + 14!
  22. E

    Advanced math subjects?

    With regards to undergraduate math courses (and beyond) there is no set, linear order. Instead, there are "branches" where different courses fall. The three main branches are: 1) Analysis: Real analysis, complex analysis, ordinary diff. eq., partial diff. eq., harmonic analysis, functional...
  23. E

    Proving trig equations using addition formulae

    First, you should have clarified in the beginning the original problem, because what you wrote: (tan(A+B)-tanA)/1+tan(A+B)tanA = tanB looks like this: \frac{\tan (A+B) - \tan A}{1} + \tan (A+B) \tan A = \tan B It looks like you changed the right side from "tan A + tan B" to "tan (A+B)". Why...
  24. E

    2 radicals in a limit?

    No. I think what HallsofIvy is trying to tell you is that you have to multiply by the conjugate twice. See my post in this thread.
  25. E

    Factors and Orders of Zeros

    OP: Is this the exact wording of the question? If so, I think it was badly worded. Is factoring "over the reals" supposed to be assumed?
  26. E

    Evaluate the limit of [(6-x)^0.5 - 2]/[(3-x)^0.5 - 1]

    This may be a big hint, but you will have to rationalize twice. First, rationalize the numerator. Simplify the numerator, but don't multiply it out in the denominator. Then rationalize the denominator. Here, you will need to multiply out two of the factors in the denominator. Eventually...
  27. E

    Solving integral from 5/(x^2 + 1) dx from -1 to 1 works one way but

    You can't choose \theta = -\frac{\pi}{4}, because by definition the range of the inverse cotangent function \theta = \cot^{-1} x is 0 < \theta < \pi. There is only one value for \cot^{-1} \left( -1 \right) and that is \frac{3\pi}{4}.
  28. E

    Trig equation

    Really? I think the above is so much simpler than your first attempt, radicals and all. I want to kick myself for not seeing it earlier.
  29. E

    Functions problem help needed

    Well, they are different. You'll have to simplify the 2nd function, but in doing so, an assumption has to be made. What is that assumption?
  30. E

    Simple trig problem

    You can simplify \sqrt{\frac{2}{18}}, you know. :tongue: Also, there's one little thing you're forgetting. MrAnchovy mentions it in his post.