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Find the slope of the curve and the equation of tangent line of the parametric equation to the given point.
5. x=√t , y = 2t + 4 , t=1
Find the first and second derivatives of the following and simplify whenever possible:
x = a cosh t; y=b sinh t
Find the first and second derivatives of the following and simplify whenever possible:
x = t ^ 2 * e ^ t y = t In t
Find the first and second derivatives of the following and simplify whenever possible:
x=e^t: y=te^-t
Find the first and second derivatives of the following and simplify whenever possible:
x = 9t ^ 2 - 1 : y = 3t+1
Find the limit of 2x(x-2)/x-2 as x is 2
x^ 2 y^ " - 2xy'-4y=x^ 2 +2 log x
Determine the critical numbers of the given function and classify each critical point
as a relative maximum, a relative minimum, or neither.
ff(tt) = tt2
tt2 + tt − 2 .
TISSUE GROWTH Suppose a particular tissue culture has area AA(tt) at time tt and a
potential maximum area MM. Based on properties of cell division, it is reasonable to assume that
the area AA grows at a rate jointly proportional to �AA(tt) and MM − AA(tt); that is
dddd
dddd = kk�AA(tt) [MM − AA(tt)]
where kk is a positive constant.
a. Let RR(tt) = AA′
(tt) be the rate of tissue growth. Show that RR′
(tt) = 0 when AA(tt) = MM/3.
b. Is the rate of tissue growth greatest or least when AA(tt) = MM/3? [Hint: Use the first
derivative test or second derivative test.]
c. Based on the given information and what you discovered in part (a), what can you say
about the graph of AA(tt)?
The derivative of a differentiable function ff(xx) is given as
ff′
(xx) = xx + 3
(xx − 2)2 .
a. Find intervals of increase and decrease for ff(xx).
b. Determine values of xx for which relative maxima and minima occurs on the graph of
ff(xx).
c. Find ff′′(xx) and determine intervals of concavity for the graph of ff(xx).
d. For what values of xx do inflection points occur on the graph of ff(xx).
