Worksheet 19
1. Sketch the following curves:
x2
(a) y = x2 +3 .
x3
(b) y = x−2 .
2. Draw a curve y = f (x) that satisfies all of the following:
• f 0 (x) > 0 when x < −1
• f has roots only at x = −2 and at x = 1
• f 00 (x) > 0 when x < −2 and when x > 2
3. For each part, design a function with the given characteristic. (You might find it easier to
design the function around x = 0 and then translate it.)
(a) The graph of f is concave up at x = 1, but x = 1 is not a local extremum.
(b) The value of f is increasing on the left of x = 1 and decreasing on the right of x = 1,
but x = 1 is not a local extremum.
(c) The derivative of f vanishes at x = 1 (meaning that f 0 (1) = 0) but x = 1 is not a local
extremum.
(d) The second derivative of f vanishes at x = 1 (meaning that f 00 (1) = 0) but x = 1 is not
a local extremum.
(e) The second derivative of f vanishes at x = 1, and x = 1 is a local extremum.
4. For each part, determine whether such a function can exist. If so, draw its graph. If not,
explain why not.
(a) f 0 (x) always negative, but f (x) never 0.
(b) f 0 (x) always positive, f 00 (x) < 0 when 2 < x < 5.
(c) f 0 (x) always positive, f 00 (x) < 0 only when 2 < x < 5.
5. Suppose f (t) is the amount of knowledge acquired after t hours of study and practice.
(a) What is the practical meaning of the inequality f 0 (2) > f 0 (1)?
(b) In practical terms, what does it mean to say that f 0 (1) > 0 and f 00 (1) > 0?
(c) In practical terms, what does it mean to say that f 0 (5) > 0 and f 00 (5) < 0?
(d) What is the practical meaning of the inequality f 0 (7) < 0?
(e) If f (t) is measured in “bits” of information, what are the units associated with f 0 (t) and
f 00 (t)?
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1. Sketch the following curves:
x2
(a) y = x2 +3 .
x3
(b) y = x−2 .
2. Draw a curve y = f (x) that satisfies all of the following:
• f 0 (x) > 0 when x < −1
• f has roots only at x = −2 and at x = 1
• f 00 (x) > 0 when x < −2 and when x > 2
3. For each part, design a function with the given characteristic. (You might find it easier to
design the function around x = 0 and then translate it.)
(a) The graph of f is concave up at x = 1, but x = 1 is not a local extremum.
(b) The value of f is increasing on the left of x = 1 and decreasing on the right of x = 1,
but x = 1 is not a local extremum.
(c) The derivative of f vanishes at x = 1 (meaning that f 0 (1) = 0) but x = 1 is not a local
extremum.
(d) The second derivative of f vanishes at x = 1 (meaning that f 00 (1) = 0) but x = 1 is not
a local extremum.
(e) The second derivative of f vanishes at x = 1, and x = 1 is a local extremum.
4. For each part, determine whether such a function can exist. If so, draw its graph. If not,
explain why not.
(a) f 0 (x) always negative, but f (x) never 0.
(b) f 0 (x) always positive, f 00 (x) < 0 when 2 < x < 5.
(c) f 0 (x) always positive, f 00 (x) < 0 only when 2 < x < 5.
5. Suppose f (t) is the amount of knowledge acquired after t hours of study and practice.
(a) What is the practical meaning of the inequality f 0 (2) > f 0 (1)?
(b) In practical terms, what does it mean to say that f 0 (1) > 0 and f 00 (1) > 0?
(c) In practical terms, what does it mean to say that f 0 (5) > 0 and f 00 (5) < 0?
(d) What is the practical meaning of the inequality f 0 (7) < 0?
(e) If f (t) is measured in “bits” of information, what are the units associated with f 0 (t) and
f 00 (t)?
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