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

Problems Plus 1. Find all functions f such that f 9 is continuous and
[ f sxd] 2 − 100 1 y x
h[ f std] 2 1 [ f 9std] 2 j dt for all real x
0
2. A student forgot the Product Rule for differentiation and made the mistake of thinking
that s ftd9 − f 9t9. However, he was lucky and got the correct answer. The function f that
he used was f sxd − e x 2 and the domain of his problem was the interval s 1 2 , `d. What was
the function t?
3. Let f be a function with the property that f s0d − 1, f 9s0d − 1, and f sa 1 bd − f sad f sbd
for all real numbers a and b. Show that f 9sxd − f sxd for all x and deduce that f sxd − e x .
4. Find all functions f that satisfy the equation
Sy f sxd dxDSy
1
f sxd dx D − 21
y
(x, y)
0
(L, 0)
FIGURE for problem 9
x
5. Find the curve y − f sxd such that f sxd > 0, f s0d − 0, f s1d − 1, and the area under the
graph of f from 0 to x is proportional to the sn 1 1dst power of f sxd.
6. A subtangent is a portion of the x-axis that lies directly beneath the segment of a tangent
line from the point of contact to the x-axis. Find the curves that pass through the point
sc, 1d and whose subtangents all have length c.
7. A peach pie is taken out of the oven at 5:00 pm. At that time it is piping hot, 100 8C. At
5:10 pm its temperature is 80 8C; at 5:20 pm it is 65 8C. What is the temperature of the
room?
8. Snow began to fall during the morning of February 2 and continued steadily into the afternoon.
At noon a snowplow began removing snow from a road at a constant rate. The plow
traveled 6 km from noon to 1 pm but only 3 km from 1 pm to 2 pm. When did the snow
begin to fall? [Hints: To get started, let t be the time measured in hours after noon; let xstd
be the distance traveled by the plow at time t; then the speed of the plow is dxydt. Let b be
the number of hours before noon that it began to snow. Find an expression for the height of
the snow at time t. Then use the given information that the rate of removal R (in m 3 yh) is
constant.]
9. A dog sees a rabbit running in a straight line across an open field and gives chase. In a rectangular
coordinate system (as shown in the figure), assume:
(i) The rabbit is at the origin and the dog is at the point sL, 0d at the instant the dog
first sees the rabbit.
(ii) The rabbit runs up the y-axis and the dog always runs straight for the rabbit.
(iii) The dog runs at the same speed as the rabbit.
(a) Show that the dog’s path is the graph of the function y − f sxd, where y satisfies the
differential equation
x d 2 y
dx
S − Î1 1 dy 2
dxD
2
(b) Determine the solution of the equation in part (a) that satisfies the initial conditions
y − y9 − 0 when x − L. [Hint: Let z − dyydx in the differential equation and solve
the resulting first-order equation to find z; then integrate z to find y.]
(c) Does the dog ever catch the rabbit?
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