Chapter 4: Q39E (page 279)
Find the critical numbers of the function.
39. \(h\left( t \right) = {t^{\frac{{\bf{3}}}{{\bf{4}}}}} - {\bf{2}}{t^{\frac{{\bf{1}}}{{\bf{4}}}}}\)
The critical numbers are \(t = 0{\rm{ and }}\frac{4}{9}\).
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For each of the numbers \(a,\,b,\,c,\,d,\,r\) and \(s\) state whether the function whose graph is shown has an absolute maximum or minimum, a local maximum or minimum, or neither a maximum nor a minimum.
Use the guidelines of this section to sketch the curve.
\(y = {x^{\bf{5}}} - {\bf{5}}x\)
Coulomb’s Law states that the force of attraction between two
charged particles is directly proportional to the product of the
charges and inversely proportional to the square of the distance between them. The figure shows particles with charge 1
located at positions 0 and 2 on a coordinate line and a particle
with charge\( - {\bf{1}}\)at a positionxbetween them. It follows from
Coulomb’s Law that the net force acting on the middle particle is
\(F\left( x \right) = - \frac{k}{{{x^2}}} + \frac{k}{{{{\left( {x - {\bf{2}}} \right)}^{\bf{2}}}}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\bf{0}} < x < {\bf{2}}\)
Where k is a positive constant. Sketch the graph of the net force function. What does the graph say about the force?
A model for the spread of a rumor is given by the equation
\(p\left( t \right) = \frac{{\bf{1}}}{{{\bf{1}} + a{e^{ - kt}}}}\)
Where \(p\left( t \right)\) is the proportion of the population that knows the rumor at time t and a and k are positive constants.
a) When will half the population have heard the rumor?
b) When is the rate of spread of the rumor greatest?
c) Sketch the graph of p.
Suppose \(f\)is a continuous function defined on a closed interval \(\left( {a,b} \right)\).
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