Math 1A – Calculus – Chapter 3 Test II - solutions.

 

1.      Consider .

a.       Simplify formulas for the first and second derivatives of  f(r).
SOLN:   and

b.      Find the inflection point for f(r).
SOLN:   where
Note that the change in concavity is barely discernable in the graph:

2.      Find equations for the tangent lines to  that are parallel to the line .
SOLN:  , so the slope of the tangent line is parallel to the given line where   Thus the equations for the tangent lines are  and  .

3.      In a fish farm, a population of the fish is introduced into a pond and harvested regularly.  A model for the rate of change of the fish population is given by the equation

where r₀ is the birth rate of the fish, P_max is the maximum population the pond can sustain and H is proportion of fish harvested in a year.  If the pond can sustain a maximum population of 5,000 fish, the birth rate is 4% and the harvesting rate is 2%, what (non-zero) population level(s) will not change, according to the model?
SOLN:  The population will not change when .  Plugging in the parameter values,  so either P = 0 or .

4.      Use the definition of the derivative to compute
SOLN:  .a

5.      Show that the curve described by the parametric equations  
has two tangent lines where x = 1/2 and find their equations.
SOLN:  x = ½  means that  where .
With k = 0, .    .  At , , so the tangent line is .   At , , so the tangent line is . 

6.      If  , find a formula for   using implicit differentiation.
SOLN: 

7.      Use logarithmic differentiation to find the derivative of
SOLN:

8.      Verify the given linearization  at a = 0.  Then determine the values of x for which the linear approximation is accurate to within 0.1.
SOLN:  Near x = 0, .  Find where the error in approximation is no more than 0.1, solve .  These numbers can be easily found using, say, a TI85 calculator, to find the roots of
y1=abs(x-ln(abs(x+1)))-0.1: