Muscle contraction. In a study of the speed of muscle contraction in frogs under various loads, British biophysicist A. W. Hill determined that the weight w (in grams) placed on the muscle and the speed of contraction v (in centimeters per second) are approximately related by an equation of the form ( w + a ) ( v + b ) = c where a , b , and c are constants. Suppose that for a certain muscle, a = 15, b = 1, and c = 90. Express v as a function of w . Find the speed of contraction if a weight of 16 g is placed on the muscle.
Muscle contraction. In a study of the speed of muscle contraction in frogs under various loads, British biophysicist A. W. Hill determined that the weight w (in grams) placed on the muscle and the speed of contraction v (in centimeters per second) are approximately related by an equation of the form ( w + a ) ( v + b ) = c where a , b , and c are constants. Suppose that for a certain muscle, a = 15, b = 1, and c = 90. Express v as a function of w . Find the speed of contraction if a weight of 16 g is placed on the muscle.
Solution Summary: The author explains how to express the speed of contraction v as a function of weight w.
Muscle contraction. In a study of the speed of muscle contraction in frogs under various loads, British biophysicist A. W. Hill determined that the weight w (in grams) placed on the muscle and the speed of contraction v (in centimeters per second) are approximately related by an equation of the form
(
w
+
a
)
(
v
+
b
)
=
c
where a, b, and c are constants. Suppose that for a certain muscle, a = 15, b = 1, and c = 90. Express v as a function of w. Find the speed of contraction if a weight of 16 g is placed on the muscle.
A driver is traveling along a straight road when a buffalo runs into the street. This driver has a reaction time of 0.75 seconds. When the driver sees the buffalo he is traveling at 44 ft/s, his car can decelerate at 2 ft/s^2 when the brakes are applied. What is the stopping distance between when the driver first saw the buffalo, to when the car stops.
Topic 2
Evaluate S
x
dx, using u-substitution. Then find the integral using
1-x2
trigonometric substitution. Discuss the results!
Topic 3
Explain what an elementary anti-derivative is. Then consider the following
ex
integrals: fed dx
x
1
Sdx
In x
Joseph Liouville proved that the first integral does not have an elementary anti-
derivative Use this fact to prove that the second integral does not have an
elementary anti-derivative. (hint: use an appropriate u-substitution!)
1. Given the vector field F(x, y, z) = -xi, verify the relation
1
V.F(0,0,0) = lim
0+ volume inside Se
ff F• Nds
SE
where SE is the surface enclosing a cube centred at the origin and having edges of length 2€. Then,
determine if the origin is sink or source.
Chapter 1 Solutions
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