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

10. If f is differentiable at a, where a . 0, evaluate the following limit in terms of f 9sad:
lim
x l a
f sxd 2 f sad
sx 2 sa
11. The figure shows a circle with radius 1 inscribed in the parabola y − x 2 . Find the center of
the circle.
y
y=≈
1
1
0
x
12. Find all values of c such that the parabolas y − 4x 2 and x − c 1 2y 2 intersect each other
at right angles.
13. How many lines are tangent to both of the circles x 2 1 y 2 − 4 and x 2 1 sy 2 3d 2 − 1?
At what points do these tangent lines touch the circles?
14. If f sxd − x 46 1 x 45 1 2
, calculate f s46d s3d. Express your answer using factorial notation:
1 1 x
n! − 1 ? 2 ? 3 ? ∙ ∙ ∙ ? sn 2 1d ? n.
15. The figure shows a rotating wheel with radius 40 cm and a connecting rod AP with length
1.2 m. The pin P slides back and forth along the x-axis as the wheel rotates counterclockwise
at a rate of 360 revolutions per minute.
(a) Find the angular velocity of the connecting rod, dydt, in radians per second,
when − y3.
(b) Express the distance x − | OP | in terms of .
(c) Find an expression for the velocity of the pin P in terms of .
y
A
O
¨
å
P(x, 0) x
16. Tangent lines T 1 and T 2 are drawn at two points P 1 and P 2 on the parabola y − x 2 and they
intersect at a point P. Another tangent line T is drawn at a point between P 1 and P 2; it
intersects T 1 at Q 1 and T 2 at Q 2. Show that
17. Show that
| | PQ1
| | | |
1 PQ2
| | − 1
PP1 PP2
d n
dx n se ax sin bxd − r n e ax sinsbx 1 nd
272
where a and b are positive numbers, r 2 − a 2 1 b 2 , and − tan 21 sbyad.
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