Math 120: Assignment 6 (Due Tue., Oct. 23 at the start of class)
Suggested practice problems (from Adams, 6th ed.):
3.2: 1, 5, 11, 13, 17, 31, 33
3.3: 1, 3, 5, 7, 13, 17, 21, 25, 29, 33, 37, 39, 43, 45, 49, 53, 57, 61,
3.4: 3, 7, 9, 13, 17, 25
Problems to hand in:
1. Compute the derivative of
(a) 3x
2
(b) e−2x ln(1/x)
2
(c) ln(et e−t )
(d) ln(ln(ln(x)))
(e) coth(x) (where coth(x) := cosh(x)/ sinh(x) = (ex + e−x )/(ex − e−x ))
(f) xx
3
2. Find an equation for the tangent to the curve xyex/y + e−1 = 0 at (1, −1).
3. In a room at temperature 20 C, your cup of (initially) ice-cold water obeys Newton’s
Law of Cooling. After 10 minutes, its temperature is 5 C. How long must you wait
until it reaches 10 C? How long must you wait until it reaches 20 C?
4. Prove that any differentiable function f (x) satisfying f 0 (x) = f (x) for all x must be
of the form f (x) = Cex for some constant C.
x
5. Find limx→0+ xx and limx→0+ xx .
6. *!* Prove that a one-to-one, continuous function on the real line must be either
increasing for all x, or decreasing for all x.
7. *!* Let f be a continuous function which is differentiable for x 6= a, and for which
limx→a f 0 (x) exists. Prove that f must also be differentiable at x = a.
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