Notation: "sqrt" means the square root function. "pi" means the number pi, 3.1416...

Collingwood, Spring 2001

1. (a) 18x² + 2     (b) (-5t²+2t + 5) e^5t / (1-t²)²     (c) 2x cos(2x³ - ln(x)) + (6x⁴ -x) sin(2x³ - ln(x))
2. (a) false     (b) false     (c) true
3. (a) -9 pi sin (3 pi t) - 27 pi sin (3 pi t) cos (3 pi t) (16 - sin²(3 pi t))^-1/2
   (b) x'(ln(t+1)) / (t+1). It would probably be a good idea to also plug in x' from part (a), to get
   [-9 pi sin (3 pi ln(t+1)) - 27 pi sin (3 pi ln(t+1)) cos (3 pi ln(t+1)) (16 - sin²(3 pi ln(t+1))^-1/2]/(t+1)
4. (a) (3x² - 2x³)/y     (b) (3/2, 3 sqrt(3)/4) and (3/2, -3 sqrt(3)/4)     (c) (0,0) and (2,0)
5. I'm going to use the character ø instead of theta for this problem, since I don't know how to make a good theta in a web page. I'll also write y for the y-coordinate of the point P.
   (a) One possible answer is: ø' = cos²(ø) y'/50.     (b) t = 10 pi/3, y = 50 sqrt(3).
   (c) When ø = pi/6, ø' = 3/20. When ø = pi/4, ø' = 1/10. So ø' is visibly not constant.

Zhang, Spring 2001

1. (a) 1/2     (b) 1
2. (a) ½ x^-1/2 e^sqrt(x) cos (e^sqrt(x))     (b) (ln t)^-2 - 2(ln t)^-3
3. y = 5x - 5
4. 1
5. There are various ways to write the answer. One is y' = (-y x^y-1 + 2y² cos x) / (x^y ln x - 4y sin x). Another way to write it is y' = (cot x - y/x) / (ln x - 2/y).

Perkins, Winter 2001

1. (a) 3e^3x / (1+e^6x)     (b) ((3+x⁴) 2 sec(2x) tan(2x) - 4x³ sec(2x)) / (3+x⁴)²     (c) (ln t)^{sin t} [cos t ln (ln t) + sin t / (t ln t)]
2. y = -x + 2
3. (a) t=0 and t=2     (b) when t=0, the velocity is 1 ft/sec; when t=2, the velocity is -1 ft/sec.
4. decreasing at 12/sqrt(10) ft/sec
5. 1/3 - 1/2700 (but I haven't covered tangent line approximations yet--they come after the midterm, at least for Sections E and G)
6. -4 pi

Leveque, Winter 2001

1. (a) (-15/4) sin(3x) (cos (3x))^1/4     (b) (30x² - 10x + 42) e^6x / (7+5x²)²     (c) tan(x) + ln(x) tan(x) + x ln(x) sec²(x)
2. 5/2 cot(5 x) + 1/2 tan(x)
3. (This is just like computing the derivative of arctan, which is done in Section 3.6 in the book.)
4. 20 ft/sec
5. (a) f'(t) is given by:

           1       if 0 < t < 2
           1       if t = 2
           5-2t    if 2 < t < 3
           0       if 3 < t < infinity
   

   The domain of f'(t) is: all positive numbers except t=3 (that is, (0,3) union (3,infinity)).
   (b)
   (c) f''(t) is given by

           0       if 0 < t < 2
           -2      if 2 < t < 3
           0       if 3 < t < infinity
   

   Its domain is: all positive numbers except t=2 and t=3.

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