Math 1A – Calculus – Chapter 3 Test II Name:______________________________
Show your work for credit. Write all responses on separate paper. Do not abuse a calculator.
1. Consider .
a. Simplify formulas for the first and second derivatives of f(r).
b.
Use calculus to find the coordinates of the inflection
point for f(r).
2.
Find equations for the tangent lines to that are parallel to the line .
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, Pmax 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?
4.
Use the definition of the derivative to compute .
5.
Show that the curve described by the parametric
equations
has two tangent lines where t = 1/3
and find their equations. Illustrate
these in a graph.
6.
If ,
find a formula for using implicit differentiation.
7.
Use logarithmic differentiation to to find the
derivative of
8. Verify the given linearization at a = 0. What interval for x for is the linear approximation accurate to within 0.1?