Math 171 — Calculus I Spring, 2016
Problem 1.
given by
Practice Questions for PDQ Exam 1
The height of the water in a tank above a hole in the tank at time t is
r
p
2
Aw g
h(t) =
h0 −
t
AT 2
where, h0 is the height of the water level above the whole at time t = 0, AT is the
cross sectional area of the tank, Aw is the cross sectional area of the whole, g is the
acceleration due to gravity and t is the time after the whole was opened.
(a) Make a rough sketch of h(t).
(b) Find a formula for the time it take for the tank to empty.
Problem 2. Let f (x) = Ax2 + Bx + C. If f (0) = 0, f (1) = 7 and f (−1) = 3, what
are the values of A, B and C? Sketch the graph of the function f (x).
Problem 3. If sinh(x) =
ex −e−x
2
find sinh(ln(10))
Problem 4. Determine whether there exists a constant c such that the line x + cy = 1
• Has slope 4
• Is horizontal
• Passes through (3, 1)
• Is vertical
And if so find the corresponding value of c for each case.
Problem 5. A hot air balloon takes off from the edge of a mountain lake. Impose
a coordinate system as show below and assume that the path the balloon follows is the
graph of the quadratic function
f (x) = −
2 2 6
x + x
2500
5
The land rises at a constant incline from the lake at the rate of 1 vertical foot for each
5 horizontal feet. Answer the following questions:
a. Find a formula for the height about the land as a function of x.
b. At what x value does the balloon hit the ground?
c. At what x value does the balloon have maximum height above the ground?
y
Lake
x
2
Problem 6. Each time the jack undergoes a complete counter clockwise revolution,
the threaded screw advances 2 millimeters upward. The distance between the threads
of a screw is called the pitch. If the tire jack has an initial height of 200 millimeters,
find the height, h(x), after x turns counter clockwise in millimeters.
Problem 7. Show that if tan(θ) = c and 0 ≤ θ ≤ π/2, then find cos(θ) and sin(θ) in
terms of the parameter c.
Problem 8. Find all angles between 0 and 2π for which sin(θ) = 1/2.
Problem 9.
Law of cosines : c2 = a2 + b2 − 2ab cos(γ) or
a2 = b2 + c2 − 2bc cos(α) or
b2 = a2 + c2 − 2ac cos(β).
The efficiency of a solar panel can be increased by ’tracking the sun’ much as the leaves
of green plants do. The solar array shown in the illustration is kept perpendicular to
the rays of sun by means of a hydraulic jack DE.
To what length S must the piston be extended to keep the array perpendicular to the
sun’s rays when the sun is at an angle of φ = 45 degrees above the horizon?
Math 171 — Calculus I Spring, 2016
Practice Questions for PDQ Exam 3
Problem 10.
x
is its own inverse by showing that f (f (x)) = x.
x−1
b. Simplify the following:
tan(cos−1 (x))
sin2 (cos−1 (x))
a. Show that f (x) =
r
2GM
R
where G is the universal gravitational constant and M is the mass of the planet. Find
the inverse of v(R), ie., expression R in terms of v. Explain what R(v) gives you.
Problem 11.
The escape velocity from a planet of radius R is v(R) =
Problem 12. An auditorium with a flat floor has a large flat panel television on one
wall. The lower edge of the television is 3 ft above the floor, and the upper edge is 10
ft above the floor. Express θ in terms of x.
Problem 13. Modern climbers use GPS to determine altitudes, but back in the day,
they used altimeters, which were based on air pressure. The basic formula is
p = p0 ekz
where p0 is the atmosperic presure at sea level, p is the atmospheric presure at the
measurement height z and the constant k depends on temperature, the acceleration due
to gravity, the universal gas constant among other things. Find a formula for the height
above sea level z in terms of air pressure p and the other parameters p0 and k.
Problem 14.
a. Solve the equation 6e4t = 3 for t.
b. Given log10 E = 4 + M , express E as a function of M .
c. The population of a city is given by (in millions) at time t (years) is P (t) = 2.1e0.1t ,
where t = 0 is the year 2000. When will the population double from its size at
t = 0?