1. See Fall 2013 exam 2:
http://www.math.colostate.edu/~calc/MATH160/exams/M160Exams_Fall2013.pdf
2. Use the definition of the derivative (see p.106-107) to find the following:
(a) f 0 (x) given f (x) = 2x2 − x + 2
(b) f 0 (x) given f (x) =
−2
x
(c) f 0 (0) given f (x) = x
p
|x|
(d) Show that f 0 (1) does not exist given f (x) = |x − 1|
3. See page 113 #27-32; page 132-134 #15-18, 21-22
4. See page 153-154 #1-18, 27-44, 48
5. See page 176 #1-60; page 147-148 #73-74
6. Understand the meaning of Theorem 1 on page 111.
7. See page 111 with regard to the derivative not existing at a point.
The following were added March 5, 2014:
1. Consider x2 + y 2 = 4.
(a) Sketch the graph.
√ √
√
√
(b) Verify that the points ( 2, 2), (− 2, − 2), (2, 0), and (0, −2) are on the graph of the curve.
(c) Find
dy
dx
(d) Sketch the lines tangent to the curve at each of these points.
(e) Determine the equation of each tangent line you drew.
√ √
√
√
(f) Are the tangent lines through ( 2, 2) and (− 2, − 2) parallel, perpendicular, or neither?
(g) Are the tangent lines through (2, 0) and (0, −2) parallel, perpendicular, or neither?
2. See note about vertical tangents on page 105 bottom.
3. Sketch the graph of a function with the following properties:
• lim f (x) = −3
x→∞
• lim− f (x) = −1
x→3
• lim f (x) = −3
• A point at which f 0 (x) does not exist, but f (x) is
continuous.
• lim− f (x) = −∞
• At least 1 local maximum
x→−∞
x→3
Note: for more sketching practice with asymptotes see page 95 #69-72
4. Find the derivative and simplify:
(a) p.139 #9,10
#9 simplifies to 0
#10 simplifies to sec2 (x)
1
(b) p.140 #12 (can simplify to − sin(x)+1
)
(c) p.140 #22 (see piazza for steps)
(d) p.140 #25 (can simplify to sec2 (θ) − csc2 (θ))
(e) y = sin(x) cot(x) (derivative is − sin(x))
The following were added March 11, 2014:
5. See page 191 #41-54, 67, 69-71
6. See page 240 #7, 8