Time and Date Stamps (logged): 12:34:21 05-21-2012 °±Ÿ²³Ÿ±°¯´Ÿ±°Ÿ±¯°± Precalculus II

Calculus I Final Exam

Completely document your work and your reasoning.

You will be graded on your documentation, your reasoning, and the correctness of your conclusions.

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Signed by Attendant, with Current Date and Time: ______________________

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Instructions:

Directions for Student:

Test Problems:

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Problem Number 1

The rate at which the position of a coasting ball on a constant incline changes is given as a function of clock time by velocity function v(t) = .00082 t^2 + .41 t + 1.2, with v in meters/sec when t is in seconds.  Determine the rate of position change for clock times t = 0, 9 and 18 sec and make a table of rate vs. clock time.

Sketch and label the trapezoidal approximation graph corresponding to this table and interpret each of the slopes and areas in terms of the situation.

Evaluate the derivative of the velocity function for t = 13.5 sec and compare with the approximation given by the graph.

By how much does the antiderivative function change between t = 0 and t = 18 seconds, what is the meaning of this change, and what is the graph's approximation to this change?

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Problem Number 2

The altitude of a sandpile is changing at the rate of 17 cm / hour.  If the volume of the sandpile is .8 h^3, where h is its altitude, then at what rate in cm^3 / hour must sand be added when its altitude is 67 cm?

 

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Problem Number 3

Determine whether f(x) = 5 sin( -2.501 / x) is continuous; if not find the points at which is it not continuous.

 

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Problem Number 4

As you zoom in toward the origin some of the functions on the following list become indistinguishable.  Group the functions so that all the functions in each group have graphs which cannot be distinguished from one another if we zoom in close enough to the origin:

y = arcsin x        y = sin x - tan x         y = x - sin x         y = arctan x         y = sin x / (1 + sin x)

y = x^2 / (1 + x^2)        y = (1 - cos x) / cos x        y = x / (1 + x^2)         y = sin x / x  - 1   

y = - x ln x         y = e^x - 1          y = x^10 / x^(1/10)    y = x / (x + 1)

For each group give the equation of the function near the origin.  code `t

 

 

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Problem Number 5

A container consists of three uniform circular cylinders stacked one on top of the other.  The first cylinder has a diameter of 16 cm, the second a diameter of 48 cm and the third a diameter of 36 cm.  The height of each cylinder is equal to its radius.  Water is being added to the system at a constant rate.

Sketch a graph showing the behavior of water depth vs. clock time.

Sketch a graph showing the behavior of the rate of depth change vs. clock time.

Sketch a graph showing the behavior of the rate at which the rate of depth change changes vs. clock time.

 

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Problem Number 6

Problem: Write a differential equation expressing the statement that the rate at which illumination changes with respect to distance x from the source is inversely proportional to the distance.

Problem: If dy / dt = .98 t^2 + .91 (t+y)/(t+1), and if at t = 0 we have y = .45, then find the approximate value of y when t = .2. Using the new values of y and t, find approximate value y when t = .4. Continue for two more steps to find the approximate value of y when t = .8.

(extra credit): Use a predictor-corrector method, with `Dt = .4  instead of the .5 used above, to find the approximate value of y when t = .8. Which value do you think is more accurate?

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Problem Number 7

Give the equations of the tangent lines to y = sin(-2.501 x) and y = 3 x at x = 0.  

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Problem Number 8

Find the derivative of   sinh( 9 cosh (-6.002 x)).

 

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Problem Number 9

If the average value of a function f(t) on the interval from t = 3 to t = 8 is found to be 11, then what is the integral int( f(t), t, 3, 8)?

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Problem Number 10

The rate at which dollars flow into your bank account is rate = 5 * 2^( .19 t), where rate is in dollars per day when t is time in months from an initial investment.   Use two 2-interval approximations to estimate the change in the value of your account between t = 9 months and t = 17 months.  One of your approximations should be an overestimate, the other an underestimate.

 

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Problem Number 11

For what interval(s) is 1 / (x^2 + 3) concave down, and for what interval(s) is it concave up?

 

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Problem Number 12

Determine whether lim{x-> 5-}( 2.5 / ( x - 5 )) exists; if so find the limit and prove its correctness.