Practice Second Midterms

Practice Second Midterm #1

50 points in all, time = 1 hour


1. (13 points) A rectangle having dimensions [Maple Math] and [Maple Math] has semicircles on three sides, as shown in the drawing below. Suppose that at first [Maple Math] and [Maple Math] , but then you want to increase [Maple Math] to 10.100 and decrease [Maple Math] in such a way that the total area of the shape (the rectangle plus the three semicircles) remains constant. Use the tangent line approximation to determine your new value for [Maple Math] .

code for diagram

[Maple Plot]

2. (12 points) Let y = [Maple Math] .

(a) Find the x - and y -coordinates of any upper and lower turning points, and use the second derivative test to determine if they are max or min.

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(b) Find all intervals of values of x where the function is increasing .

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(c) Find all vertical asymptotes (if there are any).
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(d) Find all points of inflection (if there are any), and determine the
upward or downward concavity of each part of the curve.
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(e) Sketch the curve. Show in your sketch what f(x) looks like as x approaches + [Maple Math] and - [Maple Math] , and as x approaches the vertical asymptote (if there is one) from the left and from the right.

[Maple Plot]

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code to generate graph paper

3. (10 points) Suppose that in Wanganui , New Zealand the mean daily temperature (averaged over many years) is a sinusoidal function of time. Let t be the time in months from the beginning of the year, and let y be the temperature in degrees Fahrenheit. Suppose that the hottest temperature is [Maple Math] in mid-January (that is, t = 0.5 ), and the coldest is [Maple Math] in mid-July.

(a) Graph the temperature function y = f(t) .

[Maple Plot]



(b) Give its equation, in the form [Maple Math] .

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(c) Suppose that the typical Wanganuian homeowner turns on the heat when the mean daily temperature drops below [Maple Math] . In an average year, find the time interval during the year when she has the heat turned on.

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4 . (15 points) According to the law of cosines for the triangle ABC below, we have: [Maple Math] . Suppose that [Maple Math] , [Maple Math] meters, and you are running from [Maple Math] to [Maple Math] at 5.70 m/sec. How fast is your distance [Maple Math] from [Maple Math] decreasing when you are at a distance [Maple Math] m from [Maple Math] ?

[Maple Plot]

code for diagram

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Practice Second Midterm #2

(50 points in all, time = 1 hour)


1. (13 points) Let y = [Maple Math] .

(a) Find the x - and y -coordinates of any upper and lower turning points .

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(b) Find all intervals of values of x where the function is increasing .

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(c) Find all points of inflection .
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(d) For what values of x is the function concave upward ?
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(e) Sketch the curve. Show in your sketch what f(x) does as x approaches + [Maple Math] and - [Maple Math] .


[Maple Plot]


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2. (12 points) A bicycle has a wheel whose circumference is exactly 2 meters. It is traveling at 5 m/sec from left to right. At time t = 0 it picks up a pebble in the tread of the wheel. Take (0,0) to be the point where the pebble attaches itself to the tread. In this problem use the parametric equations of the cycloid: [Maple Math] , [Maple Math] .

(a) What is the angular velocity [Maple Math] of the wheel in rad/sec?
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(b) Find formulas for the vertical velocity and the horizontal velocity of the pebble in terms of [Maple Math] .

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(c) Find the slope of the direction of motion of the pebble when t = 1/3 sec. Draw a picture of the wheel showing the position of the pebble when t = 1/3 sec, and draw an arrow showing the direction of motion of the pebble when t = 1/3 sec.


[Maple Plot]


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3. (12 points)

(a) Write the implicit equation of the ellipse whose bottom half is pictured below , and solve for x in terms of y .

[Maple Plot]

code for diagram

> p1:=plot([10*cos(t),5*sin(t),t=Pi..2*Pi] ,thickness=3):
dtline:=plot([[10*cos(Pi+Pi/6.),5*sin(Pi+Pi/6.)],
[10*cos(2*Pi-Pi/6.),5*sin(2*Pi-Pi/6)]],linestyle=3,
thickness=3):
watr:=seq([10*cos(Pi+Pi/6.+i*2*Pi/(3*20)),
5*sin(Pi+Pi/6.+i*2*Pi/(3*20))],i=0..20):
WAT:=plots[polygonplot]([watr],style=patch,
color=aquamarine):
txt:=plots[textplot]({[4,-3.5,`water`],[-.5,-1.5,`y`]},
font=[TIMES,BOLD,20]):
plots[display]({p1,dtline,WAT,txt},scaling=constrained,
xtickmarks=[-10,10],ytickmarks=[-5],
axesfont=[TIMES,BOLD,20]);


(b) Suppose that the drawing above is the side view of a container which has a circular top (and circular cross-section at any level). That is, the container is obtained by rotating the bottom half of the ellipse around the y -axis. Suppose the container is being filled with water. Let [Maple Math] be the surface area of the top of the water when it reaches the level y on the y -axis. The units are meters . Write a formula for [Maple Math] in term of [Maple Math] .
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(c) If the water is 250 cm = 2.5 m deep at its deepest point and is rising at 1 cm/sec, how fast is the surface area of the top of the water increasing?



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4. (13 points) The ellipse pictured below has foci at the two points (-5,0) and (0,-5) . Recall that the distance of any point [Maple Math] on the ellipse from (-5,0) plus its distance from (0,-5) has the constant value 20.

code for diagram

[Maple Plot]


(a) Write an equation for this ellipse in the form [Maple Math] .

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(b) Notice that the point (0,-12) is on the ellipse. Find the slope of the tangent line to the ellipse at (0,-12) .

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Practice Second Midterm #3

50 points in all, time = 1 hour


1. (12 points) Let y = f(x) = [Maple Math] .

(a) Find the x - and y -coordinates of any upper and lower turning points .

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(b) Find all points of inflection .
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(c) Sketch the curve, labeling all of the points found in parts (a) and (b) (``max,'' ``min,'' ``inflec'').
 [Maple Plot]


(d) Give the exact interval(s) where the curve is concave downward.

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2. (12 points) In the triangle shown below, the sides a and b are pieces of metal hinged at C . Here a is exactly 1.50 ,m long and b is exactly 2.50 m long. We measure the side c to be 3.50 m , but our measurement of c has an error of up to a centimeter, i.e., +/- 0.01 m . Using the law of cosines [Maple Math] , we find that the angle C is [Maple Math] .

code for diagram

[Maple Plot]

Use the tangent line approximation and implicit differentiation to determine the error in degrees (not radians) in our value for the angle C .

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3. (12 points) You are 5 ft tall, and the sun is casting a 5 ft shadow . (See the diagram.) If the sun is descending toward the horizontal at the rate of 1/4 degree per minute, how fast is your shadow lengthening? (First express the length of your shadow as a function of the sun's angle above the

horizontal.)

code for diagram


[Maple Plot]

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4. (14 points) An object oscillates horizontally at the end of a spring . Its sinusoidal horizontal motion goes from a minimum position of x = 0 to a maximum position of x = 4, m. It goes back and forth once every 2 seconds, starting at x = 0 at time t = 0 . Meanwhile, at time t = 0 the whole spring system is dropped from a height of 19.6 meters. Neglect air resistance, and take g = 9.8 [Maple Math] /s [Maple Math] .

(a) Write a formula for y(t) .

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(b) Draw a graph of x as a function of t , and find a formula for x(t) .


[Maple Plot]


(c) At the instant t = 1.25 sec, find the horizontal and vertical components of velocity , and the direction (slope) of the path.

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Practice Second Midterm #4

50 points in all, time = 1 hour



1. (15 points) Let y = f(x) = [Maple Math] ( [Maple Math] ) .


(a) Find the x - and y -coordinates of any upper and lower turning points .
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(b) Find all intervals of values of x where the function is increasing .

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(c) Find all points of inflection .
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(d) For what values of x is the function concave downward ?
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(e) Sketch the curve. Show in your sketch what [Maple Math] does as [Maple Math] approaches [Maple Math] and [Maple Math] .


[Maple Plot]

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2. (10 points) A circular oil slick of uniform thickness is caused by a spill of 1 cubic meter of oil. The thickness of the oil is decreasing at the rate of 0.1 cm/hr as the oil spreads (note: 1 cm = 0.01 m). At what rate is the radius of the slick increasing when it is 8 meters? ( Note: the volume of a circular oil slick is equal to [Maple Math] where [Maple Math] is its radius and [Maple Math] is its thickness.)

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3. (10 points) In the triangle pictured at the right, let A , B , C be the angles at the three vertices, and let a, b, c be the sides opposite those angles. According to the " law of sines ,'' you always have: [Maple Math] .

code for diagram

[Maple Plot]

Suppose that a and b are pieces of metal which are hinged at C . At first the angle A is [Maple Math] and the angle B is [Maple Math] . You then widen A to [Maple Math] , without changing the sides a and b . What happens to the angle B ? Use tangent line approximation .


4. (15 points) A conical pile of sand is exactly 10 ft high. Suppose that you have an instrument to measure the vertex angle [Maple Math] (see the diagram below).

code for cone diagram

[Maple Plot]


(a) Express the volume [Maple Math] of sand in terms of [Maple Math] .
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(b) Suppose you measure [Maple Math] to be [Maple Math] with an error of +/- [Maple Math] . Find [Maple Math] , and use the tangent line approximation to determine the error in your value for V . ( Warning : use radians for angles.)


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