Last Quarter’s Final

1. Let f(x) = {1\over{x^2}}. Compute f'(x) using the definition (the limit of a difference quotient).

2. Differentiate (find D_x(f)):
a. f(x) = 5^{3x}

b. f(x) = \log_2(x^3)

c. f(x) = \arctan({1\over x})

3. Find formulas for y' using a. logarithmic differentiation and b. the “Quotient Rule”, where = {{(2x+1)^2}\over x}.

4. Evaluate the limits (where defined; “DNE” for any that are not):
a. \lim_{x\rightarrow 3}{{x-3}\over{|x-3|}}
b. \lim_{x\rightarrow \infty} {{3x^2 + 5x - 11}\over{2x^2 + 7}}
c. \lim_{x\rightarrow 0} {{e^x-1}\over{e^{2x}-1}}

5. Find the equation of the line tangent to g(x) = ln(x) at x = 1.

6. Use implicit differentiation to find {{dy}\over{dx}} given that y = e^{xy} + x^2.

7. Find an algebraic formula for cos(arctan(x)) using a sketch of a right triangle.

8. Compute the second derivative, {{d^2 y}\over{dx^2}}, where y = \sin(x) + xe^x.

9. A balloon rises at a rate of 10 feet per second from a point on the ground 100 feet from an observer. Find the rate of change of the angle of elevation of the balloon from the observer when the balloon is 100 feet off the ground.

10. Find (both co-ordinates of) the local extrema and inflection point for the cubic function y = x^3 - 9x - 27. Sketch the curve.

11. Estimate \root3\of{123} using a tangent line approximation (hint: \root3\of{125} = 5).

12. Find the antiderivatives.
a. \int {1\over{x^2 +1}}dx
b. \int x^3 dx
c. \int dx

Exercises. Answers go in comments.
Marry, sir. Find all the typos.
Moreover. Convince a skeptic that problem 1 or one much like it is worth learning how to do (and belongs on a Calc I final) even if they know f'(x) = {{-2}\over{x^3}}. Hint: even a god-damn computer knows that; this is a course for college credit.
Secondarily. Explain why one does not go on, “Harvard Calculus” style, to require discussions about graphs (without calculations)… I thought this was a course for college credit. For extra credit, justify the lack of an “epsilon-delta” proof on this test.
Sixth and lastly. Were notations for differentiation ({{dy}\over{dx}}, f'(x), D_x (f)) just tossed around at random to confuse everyone? Or what? Justify your conclusions.
Thirdly. How in the name of the nameless name is it even remotely possible that after using TeX for seventeen freaking years, Vlorbik doesn’t know how to typeset those limits properly? For that matter, why didn’t he just scan last year’s test and upload the S.O.B.? How can one survive without being able to work the most basic tools?
Explain why, despite having been present on two or more occasions when Vlorbik explained how to do the tangent line problem on the calculator with no calculus whatsoever, students will blank out on problem 5. No, that’s easy. Explain why they think we made up a whole new symbol for the natural log when we really meant the reciprocal function… now that’s a mystery. And why should it be hard to recognize a constant function, for hecksake? What’s up with these people who haven’t learned calculus yet? Why don’t they get it?

And to conclude. Any questions?


6 Responses to “Last Quarter’s Final”

  1. Sue Says:

    Hmm, reform bashing, are we? My sense of Harvard Calculus style was, focus on concepts. (And leave out topics that aren’t essential.) Here are the questions from my final that I would’ve identified with H.C.:

    10. f prime = … (I’m too lazy right now to learn how to put that equation here, but it’s polynomial.) This tells us that f is increasing on what interval(s)?

    Extra Credit. For what value of c in the interval [0,10] will (insert definite integral, 0 to c, of polynomial here) be the largest? Explain.

    If I remember correctly, these were perhaps the hardest problems on the test. Made ’em actually think, ya know…

    Overall, my test is a bit shorter than yours, and the problems are a bit easier, I think. I stuck with polynomials more. Totally agree on giving a by definition derivative; it’s my problem #1, too.

    I’ll email it to you if you have any interest in seeing it. (word, equation editor, no tex)


  2. Sue Says:

    Now I’m embarrassed, pretty easy to write y’=x^2+3x-18… and the integral was of -x^2+4x+5.

  3. Jason Dyer Says:

    Hm, I’m not so upset about #1. I might alternatively have a diagram and a question like “explain why the formula for the definition of f'(x) can come out of this” (I realize it’s the concept that’s important here, not the calculation) but that’s likely a harder question and I’d balance including that against the rest of the test. Running through with a sample function is sort of a primitive way of testing for understanding.

    Dunno about the lack of graphs. Certainly I’d put ’em.

    I have a notion why one might want f'(x) and dy/dx (they show up enough in all the literature), but D_x(f) does seem there solely to confuse everyone.

    I feel your pain on the TeX.

  4. pat ballew Says:

    Yeah, I have one like 1) on about every third test of the year and all the major ones… even at the end of the year…

    OF course my kids would think 2 a) is Waaayy easy,,,
    just 3x(5)^(3x-1) “duh Mr. B”

    7) is sort of a prelude to figuring out that if the know the derviate of y= sin(x) then they know one for x=sin(y) …
    But how many times do you have to say “draw a triangle” (perhaps that should be in caps, sometimes I get loud)

    I always have a graph of piece-wise functions of (mostly) straight lines labled f'(x) or f”(x) and ask them to find x-values of maxima or inflection, etc… and of course the fun begins when you say that h(x) is some integral of f'(x) from a to t and ask them to find f(a) (Someone still misses it on every test)…

    And for the ultimate torture, have them take the derviative of tan^2 (x) – sec^2(x)…
    Kids??? What’s a poor math teacher to do.

  5. MLB Says:

    #11 is my new favorite problem! I just learned how to do it in class today, and it is totally cool:-) It is amazing how accurate you can be with just pencil, paper and oh yea your brain!! Who needs calculators anyway.

  6. vlorbik Says:

    wow thanks.
    “totally cool” is *exactly*
    the desired effect.

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