A sequence that arises in ecology as a model for population growth is defined by the logistics difference
equation
pn = k pn-1 (1 - pn-1 )
where pn measures the size of the population of the nth generation of a species as a proportion to its
maximum size
Let f (x) = e2 x - 3 x.
Find an equation for the tangent line to y = f (x) when x = 0.
Point: (x, y) = (0, f (0) = 0, e0 - 0 = (0, 1)
Slope:
dy
dx
x=0
= 2 e2 x - 3 x=0
= 2 e0 - 3 = -1
Equation of the tangent line:
(y - 1) = -1 (x - 0)
or
y = -x + 1
Fin
The average of a discrete number of values x1 , x2 , , xn is
range of values, f (x), a x b, is
1
b-a
1
n
nk=1 xk . The average of a continuous
b
a f (x) dx.
Find the average value of
f (x) = ex/2 , 0 x 2
The average value is
1
2-0
2 x/2
0 e
dx =
1
2
2 x
Show your work to receive credit
1. Solve the equation 5 e2 x+1 - 3 = 0 for x.
5e
2 x+1
e
(6 points)
-3=0
2 x+1
=
3
5
3
2 x + 1 = ln 5
x=
ln(3/5)-1
2
-0.755413
2. Find an equation of the tangent line to y = x e2 x+4 when x = -2.
(7 points)
Point: (x,
The Bowditch curve is defined by the parametric equations
x = sin(t /2), y = sin(t)
It is a closed curve meaning it begins to repeat itself as t gets larger and larger.
1. Starting with t = 0, determine the first t-value greater than 0 such that the curv
Evaluate both
x
x2 -4
dx
and
x3
x2 -4
dx
in two ways.
Use a trigonometric substitution.
Use the trigonometric substitution x = 2 sec() and dx = 2 sec() tan() d on both. The right triangle
that will help us get back to variable x is shown below.
x
x2 - 4
For which values of p does 2
1
x(ln(x)p
dx converge and for which does it diverge? For those that con-
verge, give the value it converges to.
Turn the improper integral into a limit
2
1
x(ln(x)p
R
dx = limR 2
Make the substitution u = ln(x),
x=R 1
up
= l
(x)
1. Find the Taylor series for cos(x) centered at 0.
The derivatives of f (x) = cos(x) are
-sin(x) if n = 1, 5, 9, 13,
-cos(x)
if n = 2, 6, 10, 14,
f (n) (x) =
sin(x) if n = 3, 7, 11, 15,
cos(x) if n = 4, 8, 12, 16,
so that the Taylor series for c
Analyze the curve y = (1/x)x .
Determine its domain.
Changing to base e,
(1/x)x = ex ln(1/x) = ex lnx = e-x ln(x) .
The domain for this function is limited only by the domain of the natural logarithm, (0, ).
-1
If its domain is (a, b), compute limxa (1