James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

y
y=≈
Q
P
0
figure for problem 4
x
3. Evaluate lim
x l 0
| 2x 2 1 | 2 | 2x 1 1 |
.
x
4. The figure shows a point P on the parabola y − x 2 and the point Q where the perpendicular
bisector of OP intersects the y-axis. As P approaches the origin along the parabola, what
happens to Q? Does it have a limiting position? If so, find it.
5. Evaluate the following limits, if they exist, where v x b denotes the greatest integer function.
v x b
(a) lim
(b) lim x v1yx b
x l 0 x
x l 0
6. Sketch the region in the plane defined by each of the following equations.
(a) v xb 2 1 v yb 2 − 1 (b) v xb 2 2 v yb 2 − 3
(c) v x 1 yb 2 − 1 (d) v xb 1 v yb − 1
7. Find all values of a such that f is continuous on R:
f sxd −H x 1 1
x 2
if x < a
if x . a
A
P
B
M
figure for problem 10
170
C
8. A fixed point of a function f is a number c in its domain such that f scd − c. (The function
doesn’t move c; it stays fixed.)
(a) Sketch the graph of a continuous function with domain f0, 1g whose range also lies in
f0, 1g. Locate a fixed point of f .
(b) Try to draw the graph of a continuous function with domain f0, 1g and range in f0, 1g
that does not have a fixed point. What is the obstacle?
(c) Use the Intermediate Value Theorem to prove that any continuous function with
domain f0, 1g and range in f0, 1g must have a fixed point.
9. If lim x l a f f sxd 1 tsxdg − 2 and lim x l a f f sxd 2 tsxdg − 1, find lim x l a f f sxd tsxdg.
10. (a) The figure shows an isosceles triangle ABC with /B − /C. The bisector of angle B
intersects the side AC at the point P. Suppose that the base BC remains fixed but the
altitude | AM | of the triangle approaches 0, so A approaches the midpoint M of BC.
What happens to P during this process? Does it have a limiting position? If so, find it.
(b) Try to sketch the path traced out by P during this process. Then find an equation of this
curve and use this equation to sketch the curve.
11. (a) If we start from 0° latitude and proceed in a westerly direction, we can let Tsxd denote
the temperature at the point x at any given time. Assuming that T is a continuous function
of x, show that at any fixed time there are at least two diametrically opposite points
on the equator that have exactly the same temperature.
(b) Does the result in part (a) hold for points lying on any circle on the earth’s surface?
(c) Does the result in part (a) hold for barometric pressure and for altitude above sea level?
12. If f is a differentiable function and tsxd − x f sxd, use the definition of a derivative to show
that t9sxd − x f 9sxd 1 f sxd.
13. Suppose f is a function that satisfies the equation
f sx 1 yd − f sxd 1 f syd 1 x 2 y 1 xy 2
for all real numbers x and y. Suppose also that
lim
x l0
f sxd
x
− 1
(a) Find f s0d. (b) Find f 9s0d. (c) Find f 9sxd.
14. Suppose f is a function with the property that | f sxd | < x 2 for all x. Show that f s0d − 0.
Then show that f 9s0d − 0.
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