M129: Applied Calculus
Q−1: [2+2+1 Marks] Let and .
a) Solve for x the equation
b) Find the points of intersection, if any, of the graphs of y = f (x) and y = 4.
c) Describe the domains of f (x) and g(x).
Q−2: [2+2+1 Marks] Let .
a) Find f (g(x)) and describe its domain.
b) Find g (f (x)), express the answer as a rational function.
c) Find the derivative of f (g(x)).
Q−3: [2+3 Marks] Find the slope of the tangent line to the following curves at the designated points:
a)
b)
Q¬−4: [2+3 Marks] Use the definition of the derivative to find f ′(x) if
a) .
b) .
Q¬−5: [2+2+1 Marks] Let .
a) Find the intervals on which the function f is increasing or decreasing and find the local extrema, if any.
b) Find the intervals on which the graph of f is concave up or concave down and find the points of inflection, if any.
c) What are the absolute maximum and minimum of f ?
Q−6: [5 Marks] Consider an open top rectangular box with a square . Find the dimensions of the box for which the volume is 32 cubic feet and the total surface area for the box is minimal.
Q¬−7: [2+3 Marks]
a) Find an equation of the tangent line to the curve at the point whose x-coordinate is 0.
b) Use the logarithmic differentiation to differentiate the function
Q¬−8: [5×1 Marks] Approximately 10,000 bacteria are placed in a culture. Let P(t) be the number of bacteria presence in the culture after t hours and suppose that P(t) satisfies the differential equation P′(t) = 0.55P(t).
a) Find the formula for P(t).
b) How many bacteria are there after 5 hours?
c) What is the growth constant?
d) What is the double-life of the bacteria in the culture?
e) How fast the bacteria are growing when they reach 100,000?
