Math 181 - Practice exam 1
The real exam will be shorter, but there will be plenty of space on the
exam paper.
Problem 1 Stride length of humans S is roughly a linear function of
height H. Given the following data points, find the equation for S in terms
of H which gives a line passing through both points.
H = 65 72
(8 marks)
S = 38 45
Problem 2 Solve the following linear system.
3a + 5b − 2c = 21
a + b + 3c = −1
a − 2b + c = 0
(8 marks)
Problem 3 Some large birds can only take flight by jumping off a cliff.
Suppose such a bird jumps at time t = 0, and as long as it has not yet started
to flap its wings, its height above water at time t is a quadratic function of
the form
h(t) = −16t2 + bt + c.
1 2.5
t(sec) =
h(feet) = 182 95
a) Tell the equations for b and c following from these data.
b) Find the equation of h(t).
c) At what time will the bird hit the water if it never starts flapping its
wings?.
d) The bird needs until t = 3 to extend its wings and start flying. Even
then, it will still go lower another 30 feet before gaining height. What is the
bird’s margin of safety (the lowest point on its path)? Note - after t = 3, the
bird’s height is no longer given by your formula for h(t). (4 + 4 + 5 + 4 =
17 marks)
Suppose we have measurements
Problem 4 Test subjects are given varying doses of a cold medicine.
Then they have to make basketball free throws for 30 seconds. The number
F of free throws is plotted against the dosis x of medicine which they received
(F on the vertical axis, x on the horizontal axis). A computer program has
determined that the regression line has equation
F = −0.245x + 3.429
a) Make a table of these data points.
b) Sketch the regression line in the plot above.
c) Find the numbers T SS and RSS of this regression model. Round
answers to 3 digits after the decimal point.
d) Find the number R2 of this regression model. Round the answer to 2
digits after the decimal point. Would you say that the regression line fits the
data well? Explain your answer very briefly.
e) Find the predicted F -value of the regression line for x = 5.
( 2 + 2 + 4 + 4 + 4 = 16 points)
Problem 5 Find the following limits.
(3+3+3=9 marks)
x−4
− 6x + 8
t2 − 4
M = lim−
t→4 t − 4
3u2 + 5u − 10, 000
N = lim
u→∞
7 + 5u
L = lim
x→4
x2
Problem 6 Suppose air pressure P (t) in an autoclave is given in PSI as
a function of time (in seconds) by
P (t) = −0.1t2 + 5t + 14.
a) Find the average rate of change of P between t = 0 and t = 3.
b) Find the instantaneous rate of change of P at t = 3.