Consider this parabola: The minimum occurs at (x,y)=(0.99,0.86), and y = 4.99 at
ID: 2126077 • Letter: C
Question
Consider this parabola: The minimum occurs at (x,y)=(0.99,0.86), and y = 4.99 at x = 1.65. Determine the rate of change of y with respect to x, at the point x = 3.20.
Let F(s)=(a2+(b2+s2)1/2)1/2 where a and b are constants.
(A) Calculate F'(a).
[Hint: F'(s) means dF/ds.]
[Data: a = 1.8; b = 1.7.]
(B) Calculate F'(b).
Consider the oscillatory function g(t)=A sin(?t), where A = 3.38 and ? =1.20. Show that d2g/dt2 = ? K g and calculate K.
Consider the oscillatory function g(t)=A cos(?t + ?) where A, ? and ? are constants.
Calculate (1/2) (dg/dt)2+(?2/2) (g)2, at some arbitrary value of t.
[Data: A=2.50, ? = 1.98, ? = 2.03.]
As a planet revolves around the sun, the distance from the sun, as a function of the angular position ?, is given by the function
r(?) = ? / [ 1+? cos(???0) ].
Calculate the perihelion and aphelion distances.
[Data: ? = 4.9, ? = 0.019, ?0 = 2.8.]
A function H(?) has the property that
[H(?)]2+[H?(?)]2 = C is constant, i.e., independent of the variable ?. Determine the function H(?), with H(0)=0. Evaluate H(?/4) if C = 1.3.
Given these parametric equations:
r(?)=?[ 1 + e cosh? ] and t(?)=T [ ?+ e sinh? ],
with ? = 1.8, T = 4.7, e = 1.6.
Calculate dr/dt as a function of r. Evaluate dr/dt at r = ?(1+e), and in the limit as r tends to infinity.
A quantity Q increases exponentially as a function of time t;
Q(t)=Q0 exp(?t).
Here Q0 is the value of the quantity at t=0; and ? = 1.1.
(A) Find the time when Q = 10 Q0.
(B) Find the time when Q = 100 Q0.
(C) Show that dQ/dt is proportional to Q, and determine the constant of proportionality.
Carbon Dating. When a plant is alive, it has a constant concentration of the isotope C-14 (compared to other isotopes of carbon) obtained from the atmosphere in carbon dioxide. But C-14 is radioactive, with half-life ? = 5730 years. After the plant dies, the concentration of C-14 decays exponentially in time,
C14(t) = C14(0) (1/2)t/?
(A) If an archeological sample of organic material has C14 = 0.01 C14(0), what is the age of the sample (i.e., the time that has passed since the material was in a living plant)?
(B) Show that dC14/dt = ? ?C14 and calculate the value of ?
A quantity q depends on a variable r. If r = 0 then q = 1. The rate of change of q(r) with respect to r is 5 q(r). Determine q(r). Calculate q(1.32).
Jack and the Beanstalk. Suppose Jack's bean produced a seedling (1 inch tall) that grew 1 inch for every inch of existing plant, per hour. How long did it take for the plant to grow from 1 inch tall to 50 feet tall?