- 1.
- Given that when
x2 - 3x + 5k is divided by x + 4 the
remainder is 9, find the value of k using Divide on the Polynomials submenu and Solve + Exact.BITMAPSETAnswer0.2171in0.2006in0ina1
- 2.
- Define functions
f (x) = x3 + x ln x and
g(x) = x + ex. Evaluate f (g(x)), g(f (x)), f (x)g(x), and f (x) + g(x).
BITMAPSETAnswer0.2171in0.2006in0ina2
- 3.
- Find the equation of the line passing through the two
points
x1, y1
,
x2, y2
.
BITMAPSETAnswer0.2171in0.2006in0ina3
- 4.
- Find the equation of the line passing through the two
points
2, 5
,
3, - 7
.BITMAPSETAnswer0.2171in0.2006in0ina4
- 5.
- Find the equation of the line passing through the two
points
1, 2
,
2, 4
.BITMAPSETAnswer0.2171in0.2006in0ina5
- 6.
- Find the slope of the line determined by the equation sx + ty = c.BITMAPSETAnswer0.2171in0.2006in0ina6
- 7.
- Find the center and semi-axes of the ellipse
x2 +4y2 + 96x - 8y + 84 = 0.BITMAPSETAnswer0.2171in0.2006in0ina7
- 8.
- Factor the difference of powers
xn - yn for
several values of n, and deduce a general formula.BITMAPSETAnswer0.2171in0.2006in0ina8
- 9.
- Applying Factor to
x2 +

- 3
x - 3
gives the factorization
showing that the system can factor some polynomials with irrational roots.
However, applying Factor to x2 - 3 and
x3 +3x2 - 5x + 1
does not do anything. Find a way to get Scientific Notebook to
factor these polynomials.BITMAPSETAnswer0.2171in0.2006in0ina9
- 10.
- If principal P is invested at an interest rate r
compounded annually, in t years it grows to an amount A given by
A = P
1 + r
. Find the number of years required for the amount
to double at interest rates of 3%, 5%, 8%, and 10%.
- Analytic solution: Solving the appropriate equation for t, define r to be each of the different interest rates, and evaluate the expression
you obtained for t.
- Experimental solution: Treat A as a function of r and t, define
different values for r and t, and make an appropriate table of values
for each interest rate r.
BITMAPSETAnswer0.2171in0.2006in0ina10
- 11.
- Find all real and complex solutions to the system of
equations
BITMAPSETAnswer0.2171in0.2006in0ina11
- 12.
- The curves
determined by quadratic equations
of
the form
Ax2 + Bxy + Cy2 + Dx + Ey + F = 0 are often referred to as conic
sections, as each can be described as the intersection of a plane with a
cone. The discriminant
of the expression
Ax2 + Bxy + Cy2 + Dx + Ey + F is the number

=
B2 - 4
AC
The type of curve determined by the quadratic equation depends on the sign of the discriminant. In nondegenerate cases, the quadratic equation
gives
where a circle is considered as a special case of an ellipse. The degenerate
cases give a pair of lines, a line, or a point, or they have imaginary
solutions.
A rotation of one or both axes in the plane leads to a change of variables
such as the change
from
x, y
to
u, v
. Show that the sign of the
discriminant (and thus the basic shape of the curve) is invariant under such
changes for a quadratic equation
Ax2 + Bxy + Cy2 + Dx + Ey + F = 0 if ad≠bc.BITMAPSETAnswer0.2171in0.2006in0ina12