# Problems Using Table Data: Int., Diff., Physics, & MVT Problems Using Table Data: Int., Diff., Physics, & MVT

Mr. Gunther Calculus BC

Name:______Date:______

In each scenario below, you’ll be given a measure of a rate of change taken at specified intervals. Your task is to answer questions about boththe level or amountand “acceleration” rates for each scenario.

1)Oil is leaking out of a tank into a water reservoir. The instantaneousrate of flow is a differentiable function and is measured every two hours over a 12-hour time period.

Time (hr) / 0 / 2 / 4 / 6 / 8 / 10 / 12
Amount (mL/sec) / 40 / 38 / 36 / 24 / 26 / 18 / 8

a)What are the units of the function for which you’ve been given values above?
Is the function a flow (i.e., a rate of change), or a level?

b)Graph the points on the function, and use any method to graphically illustrate the amount of oil spilled.

c)Give an estimate of the amount of oil that spilled in the first 4 hours, 8 hours, and 12 hours – provide the result in US gallons(1 gal ≈ 3785.41mL). Justify the technique you used to compute your results.

d)Use your results above to compute an average flowfor the first 4, 8, and 12 hours of oil leakage. What are correct units for this average flow?

e)Can you find an interval containing at least one point where the flow rate is increasing? Decreasing? Constant?What theorem lets you guarantee such a point exists.

Problems Using Table Data: Int., Diff., Physics, & MVT (pg. 2)

Mr. Gunther Calculus BC

Name:______Date:______

2)On a calm autumn evening, a firefly is hovering 1.1m above the ground, with its vertical velocity from that instant in m/min being differentiable. The following table gives values of at 1 minute intervals:

Time (mins) / 0 / 1 / 2 / 3 / 4 / 5 / 6 / 7
Vert. Vel.
(m/min) / .05 / .02 / -.02 / -.07 / -.03 / .02 / -.01 / -.02

a)At what times is the speed of the firefly .02 m/min? (This is trickier than it seems – what is speed?)

b)Use a trapezoidal approximation to estimate the fireflies height at minutes.

c)Are there any times where the firefly changes direction? Identify what you can say about when such points must occur given the table data, what property of allows you to know this, and a theorem that guarantees such points must occur.

d)Could the firefly change direction in the first minute?

e)Give an estimate of the firefly’s acceleration at 2.5 seconds and 3.5 seconds, including correct units. Do you think the firefly’s speed is increasing or decreasing at these times?

f)One of the value theorems tells you that your estimates in part e) above are more than just estimates. How can these estimates provide us with information about the actual acceleration of the fly?

g)Prove that the firefly’s acceleration must have been constant at some instant, and provide some indication of when this must have happened. (There are (at least!) two different ways to do this…)

Problems Using Table Data: Int., Diff., Physics, & MVT (pg. 3)

Mr. Gunther Calculus BC

Name:______Date:______

3)The electricity consumption of the Ricardo family’s house is measured in kilowatts – a unit of ______(power OR energy) at various points in the day, with data given in the table below – these values can be considered to come from a differentiable function .

Time / 12 mid T / 3 am / 6 am / 9 am / 12 noon / 3 pm / 6 pm / 9 pm / 12 mid W
Kilowatts / 3.9 / 4.0 / 5.1 / 4.7 / 4.3 / 4.5 / 5.3 / 4.3 / 3.8

a)What are the units of the function for which you’ve been given values above?
Is the function a flow/rate of change, or a level?
(Tricky – what units of electricity consumption measure the physical concepts of energy and power?)

b)Graph the function given and use the trapezoid method to graphically illustrate the amount of ______(fill-in: power/energy) consumed by the family.

c)Give an estimate of how much energy the house used for the day – what units will this energy usage be measured in?

d)Peak-period energy is energy that is used from 6 am – 6pm, and can be priced differently from non-peak energy. Give an estimate of the Ricardo’s peak and non-peak energy usage. Provide a precise explanation of how you will estimate each quantity.

e)Using the data above, give an estimate of how the Ricardo’s power usage is changing between 9am and 12 noon and between 12 noon and 3pm – provide correct units for this measure.

f)Is there a time when the Ricardo’s power usage is momentarily constant?

Problems Using Table Data: Int., Diff., Physics, & MVT (pg. 4)

Mr. Gunther Calculus BC

Name:______Date:______

4)A cart is attached to a spring, stretched to an initial location 10cm from its rest position, and released. Data included below give the cart’s velocity [EU1](negative values indicate leftward movement) over the first 4 seconds. You may assume these values come from a differentiable function,

Time / 0.00 / 0.50 / 1.00 / 1.50 / 2.00 / 2.50 / 3.00 / 3.50 / 4.00
Velocity
(cm/sec) / 0.00 / -9.02 / -5.29 / 5.02 / 7.70 / 0.00 / -6.93 / -4.07 / 3.86

a)Write expressions involving that would identify each of these physical concepts, at any time in the first four seconds…

• Position (assuming rest position
• Displacement from time
• Distance Traveled

b)Use the data above to provide an estimate of the position of the cart at .

c)Use the same data to carefully estimate the distance traveled by the cart at .

d)Use the data above to give an estimate of the cart’s acceleration at and . (Think carefully about what two data points above could give you the best estimate of .)

e)What does one of the value theorems say about these estimates being the actual value for ?

f)Given your finding above, justify using another value theorem that for some time . When could be?

October 13, 20151

[EU1]V(t) = -10 * (0.9)^t*sin(5pi/4*t)