11DIFFERENTIATION - Department of Mathematics

704 11 DIFFERENTIATION
In Exercises 37–40, find the given limit by evaluating
the derivative of a suitable function at
an appropriate point.
Hint: Look at the definition of the derivative.
(1 h)
37. lim
h0
3 1
h
3(2 h)
39. lim
h0
2 (2 h) 10
h
1 (1 t)
40. lim
t0
2
t(1 t) 2
x
38. lim
x1
5 1
x 1
Hint: Let h x 1.
In Exercises 41–44, find the slope and an equation
of the tangent line to the graph of the function
f at the specified point.
41. f(x) 2x2 3x 4; (2, 6)
42. f(x) 5
3 x2 2x 2;1, 5
3
43. f(x) x 4 3x 3 2x 2 x 1; (1, 0)
2
44. f(x) x 1 5
;4,
x
45. Let f(x) x 3 .
a. Find the point on the graph of f where the tangent
line is horizontal.
b. Sketch the graph of f and draw the horizontal tangent
line.
46. Let f(x) x 3 4x 2 . Find the point(s) on the graph of f
where the tangent line is horizontal.
47. Let f(x) x 3 1.
a. Find the point(s) on the graph of f where the slope
of the tangent line is equal to 12.
b. Find the equation(s) of the tangent line(s) of
part (a).
c. Sketch the graph of f showing the tangent line(s).
48. Let f(x) x 3 x 2 12x 6. Find the values of x for
which:
a. f(x) 12 b. f(x) 0
c. f(x) 12
49. Let f(x) x 4 x 3 x 2 . Find the point(s) on the graph
of f where the slope of the tangent line is equal to:
a. 2x b. 0 c. 10x
50. Astraight line perpendicular to and passing through the
point of tangency of the tangent line is called the normal
to the curve. Find an equation of the tangent line and
the normal to the curve y x3 3x 1 at the point
(2, 3).
51. G ROWTH OF A C ANCEROUS T UMOR The volume of a spherical
cancer tumor is given by the function
V(r) 4
3 r3
where r is the radius of the tumor in centimeters. Find
the rate of change in the volume of the tumor when:
a. r 2
5
cm b. r
3 4 cm
52. V ELOCITY OF B LOOD IN AN A RTERY The velocity (in centimeters
per second) of blood r centimeters from the central
axis of an artery is given by
v(r) k(R2 r2 )
where k is a constant and R is the radius of the artery
(see the accompanying figure). Suppose that k 1000
and R 0.2 cm. Find v(0.1) and v(0.1) and interpret
your results.
R
r
Blood vessel
53. E FFECT OF S TOPPING ON A VERAGE S PEED According to data
from a study by General Motors, the average speed of
your trip A (in mph) is related to the number of stops
per mile you make on the trip x by the equation
A 26.5
x0.45 Compute dA/dx for x 0.25 and x 2 and interpret
your results.
Source: General Motors
54. W ORKER E FFICIENCY An efficiency study conducted for
the Elektra Electronics Company showed that the number
of ‘‘Space Commander’’ walkie-talkies assembled by
the average worker t hr after starting work at 8 A.M. is
given by
N(t) t3 6t2 15t
a. Find the rate at which the average worker will be
assembling walkie-talkies t hr after starting work.
b. At what rate will the average worker be assembling
walkie-talkies at 10 A.M.? At 11 A.M.?
c. How many walkie-talkies will the average worker assemble
between 10 A.M. and 11 A.M.?