OLD FIRST TEST
10 pts per question, show all work for credit
Q1. Let a = h3, 4i and b = h1, 2i. Find
compa b and projb a.
Q2. A force of 3i + 2j moves an object in a straight line from the point (1, 1)
to the point (3, 2). Find the work done.
Q3. An object has position vector r(t) = t2 i + (t − 1)j. Find the average
velocity over the time interval [t, t+h] and use this to find the velocity vector
at time t.
Q4. Give an ε, δ proof that limx→5 3x − 2 = 13.
Q5. Find
√
lim ( x4 + 3x2 − x2 ).
x→∞
Q6. Find all horizontal and vertical asymptotes for
√
x2 + 4
,
y=
(x + 3)(x + 4)
and at each vertical asymptote find the left and right hand limits.
Q7. Use the definition of the derivative to find the derivative of
find the equation of the tangent line at the point (5, 2).
√
x − 1, and
Q8. Let
f (x) = x|x|.
Find the left and right hand limits as x → 0. Using the definition of continuity (which you should state), determine whether this function is continuous
at x = 0.
Q9. State the intermediate value theorem.
Show that the equation x1/3 = x + 2 has a solution.
Q10. (a) Find the derivative at x = 1 of the function
√
x+1
.
x+3
(b) Given that functions f (x) and g(x) satisfy f (0) = 1, g(0) = 1, f 0 (0) = 3
and g 0 (0) = 2, find the derivatives at x = 0 of
f (x)g(x)
.
f (x) + g(x)