Practice First Midterms

Practice First Midterm #1

(50 points in all, time = 1 hour)


1. (12 points) The graph of y = [Maple Math] is given below.

code for graph

[Maple Plot]


Find the domain of each of the following functions. You do not need to use any algebra in this problem.

(a) [Maple Math]

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(b) [Maple Math]

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(c) [Maple Math]

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(d) [Maple Math]

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2. (12 points) At time t = 0 an object starts from 0 moving to the right at 2 m/sec. Its velocity changes, as shown in the graph below. This graph shows the velocity with which the object is moving to the right as a function of time. Give the letters of ALL labeled points where.

code for diagram

[Maple Plot]

(a) the object is actually moving rightward;
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(b) the object is actually moving leftward;
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(c) the object is located to the right of the starting point;
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(d)the object is located to the left of the starting point;
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(c) the object has positive (rightward) acceleration;
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(d) the object has zero acceleration ;
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(e) the object's speed heading to the right is maximum;

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(f) the object's speed heading to the left is maximum.
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3. (13 points) A box with square top and bottom and rectangular sides has surface made up of two s by s sides and four s by t sides. Thus, if your box has dimensions 5 ft by 5 ft by 4 ft, then the total surface area A is 130 sq ft. If you want to increase s by 1 inch to 5 ft 1 in, how should [Maple Math] change so that the surface area stays the same? Use the tangent line approximation , and give your answer in feet.

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4. (13 points) A cubic container with side 9 cm holds [Maple Math] cubic cm. Suppose you have a bottle containing 750 ml of liquor . (A milliliter (ml) is the same thing as a cubic cm.) Use the tangent line approximation to find the side of a cubic container needed to hold the 750 ml of liquor.

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Practice First Midterm #2

(50 points in all, time = 1 hour)

1. (8 points) Sketch the graph of the functions

Code to generate graph paper

a. [Maple Math] ,

[Maple Plot]

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b. [Maple Math]

[Maple Plot]

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2. (6 points) The graph of the function g(x) on the interval [ [Maple Math] ] is pictured below . (This function g(x) is actually the sine function restricted to the interval [ [Maple Math] ].) Find the domain of y = [Maple Math] .

[Maple Plot]

code for diagram


3. (12 points) At time t = 0 an object starts from 0 moving to the left at 1 m/sec. Its velocity changes, as shown in the graph below. The graph below shows the velocity with which the object is moving to the right as a function of time. Give the letters of ALL labeled points where

(a) the object is actually moving rightward;
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(b) the object is actually moving leftward;

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(c) the object has positive (rightward) acceleration;
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(d) the object has zero acceleration;
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(e) the object's speed heading to the right is maximum;
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(f) the object's speed heading to the left is maximum.
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[Maple Plot]

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code for diagram

4. (12 points) From a point exactly 75 m above the ground you throw a stone upward at [Maple Math] m/sec, and you time how long it takes to hit the ground. If we neglect air resistance, the height of the stone at time t is given by the falling body formula [Maple Math] , where g = 9.8 [Maple Math] / [Maple Math] .

(a) If you time 5 sec for the stone to reach the ground, at what upward speed did you throw the stone?
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(b) If your stopwatch is accurate only to +/- 0.1 sec, how accurate is your answer to part (a)? Use the tangent line approximation.
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5. (12 points) You have a 1000-square-foot apartment which you're renting for the summer to someone from Europe , so you have to convert its area to square meters in order to explain its size. You know that there are about 10 square feet in one square meter. (More precisely, 1 square meter = 10.77 square feet). Use the tangent line approximation to find the area of your apartment in square meters. You do NOT need a calculator to do this problem correctly.
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Practice First Midterm #3

(50 points in all, time = 1 hour)

1. (8 points) The graph of y = f(x) is pictured below. It is called the " arcsine '' function, and its domain is the interval [-1,1].

Find the domain of [Maple Math] .

code for diagram

[Maple Plot]

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2. (12 points) At time t = 0 an object starts from 0 moving to the right at 1 m/sec. Its velocity changes, as shown in the graph above at the right. This graph shows the velocity with which the object is moving to the right as a function of time. Give the letters of ALL labeled points where

(a) the object is actually moving rightward;
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(b) the object is actually moving leftward;
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(c) the object has positive (rightward) acceleration;
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(d) the object has zero acceleration.

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(e) the object's speed heading to the left is maximum.

[Maple Plot]

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code for diagram

3 . (15 points) In this problem use the formula [Maple Math] for the distance an object falls in t seconds (with no air resistance). Upon arrival on a new airless planet , you decide to measure its gravitational acceleration g by dropping a stone from a height of exactly 11.25 meters and measuring the time [Maple Math] it takes to fall to the ground.

(a) If you time 3.00 sec for the stone to reach the ground, what is g?

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(b) If you do not know the time precisely, but only know that it is between 3.00 and 3.10 sec, what is the range you know g to be in? Use the tangent line approximation.
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4. (15 points) You lean a yardstick (exactly 36 inches long) against a wall, and measure the height up the wall to where the yardstick touches to be 21.6 inches pm 0.2 inches. Find the distance the other end of the yardstick is from the bottom of the wall (see the diagram below). Include the error, which you find using the tangent line approximation.

code for diagram

[Maple Plot]

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Practice First Midterm #4

50 points in all --- time = 1 hour



1. (10 points) Let [Maple Math] . Find the domain of each of the following functions:

a. [Maple Math] ,

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b. [Maple Math] .
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2. (15 points) The downward velocity of a parachutist measured in m/sec is given by the graph below.

code for diagram

[Maple Plot]

(a) Sketch the graph of downward acceleration .

[Maple Plot]

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(b) At what instant is the downward velocity the greatest? What (in practical terms) is happening at that instant? At what instant does the acceleration have the greatest positive value? What is happening at that instant? At what instant does the acceleration have the greatest negative value?

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3 . (10 points) Sketch the circle y = +/- [Maple Math] and the ellipse y = +/- [Maple Math] . The point (8,12) is on the latter ellipse. Use the tangent line approximation to find the y-coordinate of the point on the ellipse in the first quadrant whose x-coordinate is 7.7. You must use the tangent line approximation in this problem, showing your work clearly.


[Maple Plot]

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4. (15 points) A game board is to be constructed in the shape of a rectangle of dimensions r times s with a half-circle attached to each of the two sides of length r (see the figure below).

code for diagram

[Maple Plot]

(a) Find a formula for the area of the board (rectangle plus two half-circles) in terms of r and s.
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(b) You first construct a board having dimensions r = 20 cm, s = 30 cm. What is the area of the board?
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(c) Now you want to increase r to 21 cm. What must you do to s so that the total area of the board remains the same? Use the tangent line approximation. Be sure to show clearly what the function is to which you're applying the tangent line approximation formula.
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