Page 51 - 2024-calc4e-SE proofs-4e.indd

Sullivan 04 apcalc4e 45342 ch02 166 233 5pp August 7, 2023 12:54

x
Section 2.3 • The Derivative of a Polynomial Function; The Derivative of y = e and y = ln x 201
67. Tangent Lines If n is an odd positive integer, show that the (c) Assuming that the rate found in (b) remains constant, how
n
tangent lines to the graph of y = x at (1, 1) and at (−1, −1) are much would the luminosity change if its photosphere
parallel. temperature increased by 1 K (1 C or 1.8 F)? Compare this
◦
◦
68. Tangent Line If the line 3x − 4y = 0 is tangent to the graph change to the present luminosity of the Sun.
3
of y = x + k in the first quadrant, find k.
76. Medicine: Poiseuille’s Equation The French physician
69. Tangent Line Find the constants a, b, and c so that the graph Poiseuille discovered that the volume V of blood (in cubic
2
of y = ax + bx + c contains the point (−1, 1) and is tangent to centimeters per unit time) flowing through an artery with inner
© 2024 BFW Publishers PAGES NOT FINAL - For Review Purposes Only - Do Not Copy.
the line y = 2x at (0, 0). radius R (in centimeters) can be modeled by
70. Tangent Line Let T be the tangent line to the graph of y = x 3
4
V (R) = kR
1 1
at the point , . At what other point Q on the graph
2 8 π
where k = is constant (here ν represents the viscosity of
3
of y = x does the line T intersect the graph? What is the slope 8νl
of the tangent line at Q? blood and l is the length of the artery).
71. Military Tactics A dive bomber is flying from right to left (a) Find the rate of change of the volume V of blood flowing
2
along the graph of y = x . When a rocket bomb is released, it through the artery with respect to the radius R.
follows a path that is approximately along the tangent line. Where (b) Find the rate of change when R = 0.03 and
should the pilot release the bomb if the target is at (1, 0)?
when R = 0.04.
72. Military Tactics Answer the question in Problem 71 if the
3
plane is flying from right to left along the graph of y = x . (c) If the radius of a partially clogged artery is increased
from 0.03 to 0.04 cm, estimate the effect on the rate of
73. Fluid Dynamics The velocity v of a liquid flowing through change of the volume V with respect to R of the blood
a cylindrical tube is given by the Hagen–Poiseuille flowing through the enlarged artery.
2
2
equation v = k(R −r ), where R is the radius of the tube, k is a
constant that depends on the length of the tube and the velocity of (d) How do you interpret the results found in (b) and (c)?
the liquid at its ends, and r is the variable distance of the liquid
from the center of the tube. See the figure below. 77. Derivative of an Area y
Let f (x) = mx, m > 0. f(x)  mx
(a) Find the rate of change of v with respect to r at the center of
Let F(x), x > 0, be defined as
the tube.
the area of the shaded region in
(b) What is the rate of change halfway from the center to the wall the figure. Find F (x).
′
of the tube? x x
(c) What is the rate of change at the wall of the tube? 78. The Difference Rule Prove that if f and g are differentiable
functions and if F(x) = f (x) − g(x), then
r R
′
′
′
F (x) = f (x) − g (x)
v v
0 t
n
79. Simple Power Rule Let f (x) = x , where n is a positive
74. Rate of Change Water is leaking out of a swimming pool that integer. Use a factoring principle to show that
measures 20 ft by 40 ft by 6 ft. The amount of water in the pool at
2
a time t is W(t) = 35,000 − 20t gallons, where t equals the ′ f (x) − f (c) n − 1
number of hours since the pool was last filled. At what rate is the f (c) = lim x − c = nc
x→c
water leaking when t = 2 h?
80. Normal Lines For what nonnegative number b is the line given
75. Luminosity of the Sun The luminosity L of a star is the rate at
1
3
which it radiates energy. This rate depends on the temperature T by y = − x + b normal to the graph of y = x ?
and surface area A of the star’s photosphere (the gaseous 3
2
surface that emits the light). Luminosity is modeled by the 81. Normal Lines Let N be the normal line to the graph of y = x at
4
equation L = σ AT , where σ is a constant known as the the point (−2, 4). At what other point Q does N meet the graph?
Stefan–Boltzmann constant, and T is expressed in the absolute
(Kelvin) scale for which 0 K is absolute zero. As with most stars, Challenge Problems
the Sun’s temperature has gradually increased over the 6 billion
years of its existence, causing its luminosity to slowly increase. 82. Tangent Line Find a, b, c, d so that the tangent line to
2
3
the graph of the cubic y = ax + bx + cx + d at the point (1, 0)
(a) Find the rate at which the Sun’s luminosity changes with is y = 3x − 3 and at the point (2, 9) is y = 18x − 27.
respect to the temperature of its photosphere. Assume that the
83. Tangent Line Find the fourth degree polynomial that
surface area A remains constant.
contains the origin and to which the line x + 2y = 14 is tangent at
(b) Find the rate of change at the present time. The temperature both x = 4 and x = −2.
◦
of the photosphere is currently 5800 K (10,000 F), the 84. Tangent Lines Find equations for all the lines
8
radius of the photosphere is r = 6.96 × 10 m, containing the point (1, 4) that are tangent to the graph
W of y = x − 10x + 6x − 2. At what points do each of the
3
2
and σ = 5.67 × 10 −8 .
2
m K 4 tangent lines touch the graph?
© 2024 BFW Publishers PAGES NOT FINAL
For Review Purposes Only, all other uses prohibited
Do Not Copy or Post in Any Form.