MATHEMATICS 184 (Section 922) - TERM EXAM 4
NAME:
STUDENT ID NUMBER:
SIGNATURE:
INSTRUCTIONS: No notes or books are to be used. Calculators
are allowed. No credit will be given for the correct answer without
the (correct) accompanying work. Use the back of the pages if you
need extra space.
1.(a) Find the local linearization of √
the function
3
f (x) = x
near x = -8
[3]
(b) Calculate
[3]
lim
x→∞
ln(x)
√
3
x
(c) Calculate
[3]
lim
x→0
sin(x)
ex
2. When you cough, your windpipe contracts. The speed, v, with
which air comes out depends on the radius, r, of your windpipe. If
R is the normal (rest) radius of your windpipe, then for r ≤ R, the
speed is given by:
v = a(R − r)r 2
where a is a positive constant. What value of r maximizes the
speed?
[6]
3. Find an equation of the tangent line to the curve
2(x2 + y 2 )2 = 25(x2 − y 2 )
at the point (3, 1).
[10]
4. For the function
f (x) = x2 ex
(a) Identify the domain and the range.
[2]
(b) Find the coordinates of the x and y-intercepts.
[2]
(c) Find all the critical points and corresponding critical values.
[4]
(d) What are the coordinates of the local maxima and minima?
(Specify which ones are local maxima and which are local minima).
[2]
(e) Find the coordinates of any inflection points.
[4]
(f) Using the information from parts (a)-(e), sketch a graph of this
function.
Warning: Your sketch must match the information you determined above. No marks will be awarded for a correct sketch
that does not match your results from (a)-(e).
[3]
5. Differentiate the following functions:
(a) f (x) =
ln(x)
x
[2]
(b) y = (sin(x)cos(x))99
2
(c) y = arctan(ex + 7)
(Simplify your answer).
[3]
[3]