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

Problems Plus 1. If a rectangle has its base on the x-axis and two vertices on the curve y − e 2x 2 , show that
the rectangle has the largest possible area when the two vertices are at the points of inflection
of the curve.
2. Show that | sin x 2 cos x | < s2 for all x.
3. Does the function f sxd − e 10| x22|2x 2 have an absolute maximum? If so, find it. What about
an absolute minimum?
4. Show that x 2 y 2 s4 2 x 2 ds4 2 y 2 d < 16 for all numbers x and y such that | x | < 2 and
| y | < 2.
5. Show that the inflection points of the curve y − ssin xdyx lie on the curve y 2 sx 4 1 4d − 4.
6. Find the point on the parabola y − 1 2 x 2 at which the tangent line cuts from the first
quadrant the triangle with the smallest area.
7. If a, b, c, and d are constants such that
ax 2 1 sin bx 1 sin cx 1 sin dx
lim
− 8
x l 0 3x 2 1 5x 4 1 7x 6
Q
y
find the value of the sum a 1 b 1 c 1 d.
8. Evaluate
sx 1 2d 1yx 2 x 1yx
lim
xl` sx 1 3d 1yx 2 x 1yx
0 x
FIGURE for PROBLEM 11
P
9. Find the highest and lowest points on the curve x 2 1 xy 1 y 2 − 12.
10. Sketch the set of all points sx, yd such that | x 1 y | < e x .
11. If Psa, a 2 d is any point on the parabola y − x 2 , except for the origin, let Q be the point
where the normal line at P intersects the parabola again (see the figure).
(a) Show that the y-coordinate of Q is smallest when a − 1ys2 .
(b) Show that the line segment PQ has the shortest possible length when a − 1ys2 .
FIGURE for PROBLEM 13
y
y=≈
B
A
y=mx+b
O P
FIGURE for PROBLEM 15
x
12. For what values of c does the curve y − cx 3 1 e x have inflection points?
13. An isosceles triangle is circumscribed about the unit circle so that the equal sides meet
at the point s0, ad on the y-axis (see the figure). Find the value of a that minimizes the
lengths of the equal sides. (You may be surprised that the result does not give an equilateral
triangle.).
14. Sketch the region in the plane consisting of all points sx, yd such that
2xy < | x 2 y | < x 2 1 y 2
15. The line y − mx 1 b intersects the parabola y − x 2 in points A and B. (See the figure.)
Find the point P on the arc AOB of the parabola that maximizes the area of the
triangle PAB.
16. ABCD is a square piece of paper with sides of length 1 m. A quarter-circle is drawn from
B to D with center A. The piece of paper is folded along EF, with E on AB and F on AD,
so that A falls on the quarter-circle. Determine the maximum and minimum areas that the
triangle AEF can have.
17. For which positive numbers a does the curve y − a x intersect the line y − x?
18. For what value of a is the following equation true?
lim
x l `S x 1 a x
x 2 aD
− e
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