Linear Algebra + Basic Calculus

Linear Algebra + Basic Calculus
Show all working; problems that do not show their work will typically receive reduced or zero
marks. Late assignments cannot be marked under any circumstances.
1. (Cross products and their applications.)
(a) Find the cross product (1; 1; 1) (1; 2; 3).
(b) Using this cross product, describe the plane through the point (4; 5; 6) parallel to the vectors
(1; 1; 1) and (1; 2; 3) using the point-normal equation.
2. (Determinants and their applications.) Consider the two matrices
A =
1 2 3 4
5 6 7 8
9 10 11 12
13 14 15 16
; B =
1 2 2
2 1 2
2 2 1
(a) Find the determinant of A.
(b) Does A??1 exist? Explain why.
(c) Find the determinant of B.
(d) Does B??1 exist? Explain why.
3. (Derivatives.) Calculate the derivatives of the following expressions with respect to x. Make sure
to show your work. Simplify your answers when possible.
(a) y = x3 + 3×2 + ex + ln(x)
(b) y = (cos(x) sin(x))3
(c) y =
cos2(x + ) + sin2(x ?? )
(d) y = cos(cos(cos(cos(x))))
4. (Implicit dierentiation.) Use implicit dierentiation to calculate
for each of the following
equations. Again, show all of your work, and simplify your answers when possible.
(a) x4 + 4x3y + 6x2y2 + 4xy3 + y4 = 1: (b) ex+y = ln(y):
5. (Calculation: curve sketching.) Consider the function f : R ! R dened by setting f(x) = 2×3 +
5×2 ?? 4x ?? 3.
(a) Determine the intervals on which f is increasing and the intervals on which f is decreasing, as
well as the intervals on which f is concave up and the intervals on which f is concave down.
(b) What are the relative maxima of f(x)? What are the relative minima of f(x)?
(c) Draw by hand the graph of f(x) over the interval [??4; 2].
6. (Matrix inverses and their applications.) Congratulations! You’ve been hired to help the city-state
of Lancre manage its economy. Lancre has three things that it makes: sheep (S), grain (G), and
workers (W). To make more sheep, however, you need to have some sheep to start with, along with
grain to feed them and workers to tend to them. Similarly, to make grain you need seeds to start
with and workers to plant the seeds, and to have workers you need to feed them sheep and grain
and train them with other workers.
After some research, you’ve determined that the following equations model sheep, grain, and workers:
if you want to have Cs surplus sheep, Cg surplus grain, and Cw surplus workers at the end of the
year, you want to produce S sheep, G grain and W workers in total so that the three equations
3S ?? 4G ?? 1W = Cs
??1S + 3G ?? 1W = Cg
??1S ?? 1G + 2W = Cw
are all satised.
(a) Write out the coecient matrix A corresponding to this system of linear equations.
(b) Calculate A??1.
(c) What should S; G;W should be if the Queen of Lancre wants to export one sheep, import two
workers, and export one grain? (I.e. what should S; G;W be if Cs = 1;Cg = ??2;Cw = 1?)
(d) Same as above, but for Cs = 0;Cg = ??1;Cw = 1.