CALCULUS 1000A – Section 570
Problem Set # 4
Due: November 16th at 11:59PM (London Time) on gradescope.ca
For full credit, ensure you show all your work and explain your reasoning. Simply stating
the final answer for any of the following questions will not result in full credit. Each question
has one point awarded for communication of mathematical ideas, in addition to the points
awarded for your solution. Your solution should be well written and concise.
You are welcome to work with one (1) other peer in our section of calculus, and submit one
problem set as a pair. Please ensure when submitting to Gradescope that you have identified
both yourself and your partner. Failure to identify both students may result in the loss of
points.
Total Points for this Problem Set: 35 points
1. (5 points + 1 point for comm.) Consider the function
f(x) = x
sin(x)+2
.
What is the equation of the tangent line that passes through x =
π
2
? Ensure to show
all your work for full points.
2. (4 points + 1 point for comm.) When the price of dog collars is p dollars per collar,
the manufacturer is willing to supply x hundreds of collars to market. The supply is
determined through the equation
x
2 − 8x
√
p − 2p
2 = 48 .
How fast is supply changing when the price is $4.00 per collar and is increasing at
$0.10 per week?
3. (+ 1 point for comm.) A water tank has the shape of an inverted circular cone (meaning
the point of the cone is down) with a base radius of 6 metres and a depth of 8 metres.
Suppose that water is being pumped into the tank at a constant rate of 4 cubic metres
per minute.
(a) (2 points) Draw a picture of the conical tank, including a sketch of the water
level at a point in time when the tank is not yet full. Introduce variables that
measure the radius of the water’s surface and the water’s depth in the tank, and
label them on your figure.
(b) (1 point) What equation relates the radius and the depth of the water at time t,
and why?
1
(c) (1 point) What equation relates the volume of the water in the tank at time t to
the depth of the water at time t?
(d) (2 points) Through differentiation, find an equation that relates the rate at which
the volume is changing to the rate at which the water depth is changing.
(e) (2 points) When is the water depth increasing most rapidly: at h = 3 or h = 5?
Explain your reasoning.
4. (+ 1 for comm.) Suppose that h(x) is a differentiable function such that h
′
(2) = 0.
Suppose further that on the intervals 1 < x < 2 and 2 < x < 3 it is known that h ′ (x) is positive. (a) (2 points) Does h(x) have a local minimum, local maximum or neither at x = 2? Briefly explain. (b) (1 point) Suppose that h ′′(x) exists for all x in the interval 1 < x < 3. Reasoning graphically, describe the behaviour of h ′′(x) near x = 2. Meaning, what is occurring to the sign of h ′′(x) near x = 2? (c) (1 point) Other than being a critical point for the function h(x), what else is special about x = 2 in terms of behaviour of h(x)? 5. (+ 1 for comm.) Consider the function f(x) = x x 2 + 4 , with first and second derivatives f ′ (x) = 4 − x 2 (x 2 + 4)2 and f ′′(x) = 2x 3 − 24x (x 2 + 4)3 . (a) (2 points) Determine all vertical and horizontal asymptotes. (b) (3 points) Determine the intervals on which the function is increasing and decreasing. Classify all critical points as either local minimums or maximums. (c) (3 points) Determine the intervals on which the function is concave up or down, and any inflection points. (d) (1 point) Sketch the graph of f(x)