M ATHEMATICS 53: Elementary Analysis I FINAL EXAM
INSTRUCTIONS: Write all necessary solutions on your bluebooks and box all final answers. Calculators are not allowed. You have two hours to finish this exam. Good luck! I.
∫ ∫ √ ∫ √ √
True or False. Write T if the statement is always true. Otherwise, write F. (1 ( 1 point each) each ) 1. The function has a horizontal tangent line at the point (2, -16). 2. If , then f then f has has an inflection point at x = x =a. 3. The graph of the function is symmetric with respect to the y the y -axis. -axis. 4. If f If f is is differentiable on [a [a, b], then f then f is is integrable on [a [a, b]. 5. The function
is defined for all x all x < < 5.
II. Consider
Discuss the continuity of f on f on [0,1]. (5 (5 points) III. Solve for 1.
. Do not simplify. (4 ( 4 points) points)
2.
IV. Find the limits. (4 (4 points) points) 1.
2.
V. Evaluate the following integrals. (5 (5 points) points) 1. 2.
VI. Consider the region enclosed by the curves
3.
and
shown on the right. Set-up the integrals representing the following:
. A sketch of the graph is
1. The perimeter of the region. (3 points) 2. The total area of the regions R1 and R2. (4 points)
√
3. The volume of the solid generated when R 1 is rotated about the line
using the method of
Washers. (4 points) 4. The volume of the solid generated when R2 is rotated about the line Cylindrical Shells. (4 points)
using the method of
VII.Solve the following problems completely.
√
1. Find the equation of the line normal to the curve
at the point where
. (4 points) 2. A particle moves along the curve so that its abscissa is increasing at a rate of 2 units per second. At what rate is the particle moving away from the origin as it passes through the point (e, 1)? (5 points) 3. Find the area of the largest rectangle that can be inscribed in the region bounded by the curve , the line and the positive x -axis. (5 points)
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TOTAL NUMBER OF POINTS : 70
M ATHEMATICS 53: Elementary Analysis I FINAL EXAM
Directions: This exam is good for two hours only. Show all complete and neat solutions. I.
| | √ √ ∫ ∫∫ √ √
Let
Discuss and indicate the type of discontinuity (if any) of the function at the points x = 0 and x = 1. (5 points) II. Given
and its derivatives
and
, the following table can be
obtained:
1. Where are the critical points(s) and the point(s) of inflection of f ? ( 2 points) 2. In what interval(s) is f increasing or decreasing? ( 2 points) 3. In what interval(s) is the graph of f concave upward or concave downward? ( 2 points) 4. Where are the relative extrema? ( 2 points) 5. Using limits, find the vertical and horizontal asymptotes of the graph of f . ( 2 points) 6. Sketch the graph of . ( 2 points)
Interval
0
0
0
undefined
d.n.e.
d.n.e.
III. Evaluate the following limits. (4 points each) 1.
IV. Find 1.
2.
. Do not simplify. (5 points each)
2.
(use logarithmic diff.)
V. Evaluate the following integrals. (5 points each) 1. 2.
VI. Let R be the region bounded by will give:
3.
,
, and
. Set up the definite integrals that
1. the perimeter of R. (3 points) 2. the area of R. (3 points) 3. the volume of the solid generated by revolving R about the line x = -1 using Cylindrical Shells. (4 points)
VII.Solve the following problems completely. (5 points each) 1. Find the points on the graph that are closest to the point (0,1). 2. A ladder is 25 feet long and leaning against a vertical wall. The bottom of the ladder is being pulled horizontally way from the wall at 3 feet per second. Determine how fast the top of the ladder is sliding down the wall when the bottom is 15 feet from the wall. 3. A ball is thrown downward from a window that is 80 feet above the ground with an initial velocity of -64 feet per second. a. Find the distance function, , in terms of time t (in seconds). Given that the acceleration due to gravity is
.
b. How long will it take the ball to reach the ground?
---END OF EXAM---
TOTAL NUMBER OF POINTS : 75
M ATHEMATICS 53: Elementary Analysis I FINAL EXAM
Directions: Show all necessary solution steps and box all final answers. I.
(√ ) [ ] √
Evaluate the following limits. (5 points each) 1.
2.
3.
II. Differentiate the following to solve for 1.
. No need to simplify. (5 points each) 2.
III. Given that
.
1. Using limits, determine the equations of the horizontal and vertical asymptotes of the graph of f. ( 2 points) 2. Find the intervals where f is increasing or decreasing. ( 2 points) 3. Find the intervals where the graph of g is concave up or concave down. ( 2 points) 4. Find the coordinates of the critical and inflection points of the graph of f. ( 2 points) 5. Sketch the graph of f , with emphasis on concavity. ( 2 points)
∫∫ √ √
∫∫ √
IV. Evaluate the following integrals. (6 points each) 1. 2.
V. Let R be the region bounded by the curves integral that will yield the following:
3. 4.
,
, and
. Setup the definite
1. The area of area of R using vertical strips. ( 2 points) 2. The perimeter of R, in terms of y . (3 points) 3. The volume of the solid obtained by revolving R about the line x = 2, by the method of Cylindrical Shells. (3 points) 4. The volume of the solid obtained by revolving R about the line x = -3, by the method of Washers. (3 points) VI. Answer completely the following word problems. (5 points each)
√
1. Find the height of a right circular cone, whose slating side is feet long, having the largest possible volume. 2. A light is hung 15 feet atop a lamp post on a level ground. If a man 6 feet tall is walking away from the light at a rate of 6 feet per second, how fast is his shadow on the ground lengthening? ---END OF EXAM---
TOTAL NUMBER OF POINTS : 80
M ATHEMATICS 53: Elementary Analysis I FINAL EXAM
General Directions: This exam is good only for two hours. Do as indicated. Write your answers on your bluebook, clearly and legibly. Show all necessary solutions and box your final answers. Use balck or blue ink only. Insert the questionnaire on your bluebook after the exam. Good luck! I.
√
Evaluate the following limits. (5 points each) 1. 2.
II. Given:
3.
Discuss the continuity at and determine if it is continuous or discontinuous at those points. Classify each discontinuities whether it is essential or removable. (9 points)
III. Find 1.
. (5 points each)
2.
IV. Consider the graph of the equation: 1. Verify: 2. 3. 4. 5. 6.
and
. ( 2 points)
Using limits, determine all the equations of the asymptotes of the graph. ( 2 points) Find all the intervals for which the graph of f is increasing/decreasing. ( 2 points) Find all the intervals for which the graph of f is concave upward/downward. ( 2 points) Give the coordinates of the relative extrema and point(s) of inflection. ( 2 points) Sketch the graph of f , with emphasis on concavity. ( 2 points)
∫ √
V. Integrate. (5 points each) 1.
2.
∫ √
VI. Let R be the region bounded by the curves will yield the following:
3.
and
∫ √
. Set-up the integrals that
1. The area of R using horizontal strips. ( 2 points) 2. The volume of the solid obtained by revolving R about the line using the method of Washers. (3 points) 3. The volume of the solid obtained by revolving R about the y -axis using the method of Cylindrical Shells. (3 points) 4. The perimeter of R, in terms of x . (3 points) VII.Solve the problems completely. (4 points each) 1. A box with a square base and open top must have a volume of 32 cubic feet. Using derivatives, find the dimensions of the box that will minimize the amount of material to be used. 2. Air being pumped into a spherical balloon. Find the rate of increase in the radius of the balloon at the time when its radius is 10 inches and its volume is increasing at a rate of 400 cubic inches per second at that time. ---END OF EXAM---
TOTAL NUMBER OF POINTS : 80
M ATHEMATICS 53: Elementary Analysis I FINAL EXAM
Directions: Write all necessary solutions on your bluebooks and box all final answers. You have two hours to finish this exam. Good luck! I.
|| √
Evaluate the following limits. (4 points each) 1.
2.
II. Discuss the continuity of the function
at
.
III. Find 1. 2.
. Do not oversimplify. (4 points each)
IV. Consider the function 1. 2. 3. 4. 5.
, with
, and
.
Using limits, determine the asymptotes of the graph of f . (4 points) Find the intervals on which the graphs of f is increasing or decreasing. (3 points) Find the intervals on which the graph of f is concave up or concave down. (3 points) Give the coordinates of the points where f has a relative extremum or point of inflection. ( 2 points) Sketch the graph of f and label the relative extrema, inflection point and asymptotes. (3 points)
∫∫ √ ∫
V. Find the following anti-derivatives and evaluate the definite integral. (4 points each) 1. 2. 3.
VI. Let R be the region bounded by following.
,
, and
. Set-up the integrals representing the
1. The area of R using horizontal strips. (3 points) 2. The perimeter of R. (4 points) 3. The volume of the solid generated when R is revolved about the line a. Washers. (4 points) b. Cylindrical Shells. (4 points)
using the method of:
VII.Solve the following problems completely. 1. Use an appropriate approximation to estimate the value of . (3 points) 2. Given , find the value(s) of where the slope of the tangent line to curve is minimum. (5 points) 3. Jello is walking east away from an intersection at a speed of 1 feet per second while Chris is running south towards the intersection at some constant speed c. At the instant when Jello and Chris are 30 and 40 feet away, respectively, from the intersection, the distance between them is decreasing at a rate of 5 feet per second. Find the value of c.
---END OF EXAM---
TOTAL NUMBER OF POINTS : 80
M ATHEMATICS 53: Elementary Analysis I FINAL EXAM
Directions: This exam is for two hours. Use only black or blue pen. Show all solutions and box your final answer. I.
√
Find 1. 2.
. No need to simplify. (5 points each)
II. Evaluate the following limits. (4 points each) 1.
2.
3.
| | ∫∫ √ ()√ √ ) ( ∫ √ ∫
III. Given:
Discuss the continuity of f at
and
. Identify each type of discontinuity, if any.
IV. Find the following antiderivatives. (5 points each) 1. 2. 3.
V. Evaluate the definite integral:
. (5 points)
VI. Word Problems. (5 points each) 1. Determine the interval/s on which the graph of is concave downward. 2. A car driver travelling at a constant velocity of 90 feet per second along a straight road applies the brake such that the car has a constant negative acceleration of 15 feet per second squared. a. Find an expression for where s is in feet measured from the point where the brake was first applied . b. How long will it take for the car to come to a stop? c. How far will the car travel before stopping? 3. A piece of wire 80 centimeters long is bent to form a rectangle. Find the dimensions of the rectangle so that its area is as large as possible. 4. Suppose that John’s tumor is spherical in shape and its radius is decreasing at the rate of
centimeters per day. What is the rate of decrease of the volume of his tumor when its radius is 0.4 centimeters?
VII.Let R be the region bounded by the graphs of integral equal to
,
, and the x -axis. Set-up the definite
1. The area of R using horizontal strips. (3 points) 2. The volume of the solid generated when R is revolved about the line with equation method of Disks. (3 points) 3. The volume of the solid generated when R is revolved about the line with equation method of Cylindrical Shells. (3 points) 4. The perimeter of the boundary of R, in terms of x . (3 points) ---END OF EXAM---
TOTAL NUMBER OF POINTS : 80
using the using the
M ATHEMATICS 53: Elementary Analysis I FINAL EXAM
Directions: This exam is for two hours only. Use blue or black pen only. Show all solutions. I.
Evaluate the following limits. (4 points each) 1. 2. 3.
√
II. Given
Determine whether the function is continuous at the type of discontinuity. (9 points) III. Find 1.
. If continuous, identify
. Do not simplify. (5 points each)
,
, and
.
Find the x - and - and y -intercepts of the graph of f . ( 2 points) Determine the intervals when the function is increasing/decreasing. (3 points) Determine the intervals when the graph of the function is concave up/down. (3 points) Find the coordinates of the relative extrema and inflection points of the graph of f . (4 points) Sketch the graph of f . (4 points)
∫
∫
V. Evaluate the following integrals. (4 points each) 1.
, and
2.
IV. Given 1. 2. 3. 4. 5.
,
2.
∫ √ √
VI. Let R be the shaded region bounded by the graphs of the definite integrals to find the following. (3 points each)
3.
,
, and the x -axis. Set-up
1. Perimeter of the region in terms of the variable x . 2. Area of the region using vertical rectangular strips. 3. Volume of the solid, generated by revolving the region about the line Shells. VII.Solve the following problems systematically.
, using Cylindrical
√
1. Use local linear approximation to estimate the value of . (3 points) 2. Find two negative real numbers whose sum is the biggest if their product is 8. (3 points) 3. Mark and Diane are jogging at constant speeds along perpendicular roads, with Mark travelling toward the intersection of the two roads at 10 feet per second. If at the instant when Mark and Diane are 5 and 12 feet away from the intersection, respectively, the distance between them is decreasing at a rate of 2 feet per second, at what speed and in what direction is Diane jogging? ( 5 points) 4. A particle observed to be moving along the x -axis has its velocity described by , where t is in seconds and v is in meters per second. In addition, the particle was found to be 45 meters to the right of the origin at seconds. a. Find its acceleration at the instant when the velocity was meters per second. (3 points) b. Determine the function which describes the position of the function with respect to the origin at any time t . (3 points)
---END OF EXAM---
TOTAL NUMBER OF POINTS : 80
M ATHEMATICS 53: Elementary Analysis I FINAL EXAM
Directions: This exam is for two hours only. Write your answers in blue books using black or blue ink only. Show all necessary solutions and box all final answers. I.
√
Evaluate the following limits. (4 points each) 1.
2.
√ ∫ () √ √ ∫ ∫ √
II. Let f be the function defined by
Test for continuity at removable. (6 points)
III. Find
and at
. Classify each discontinuity as either essential or
. There is no need to simplify. (5 points each)
1.
2.
IV. Given
and the following table describes the sign of
and
on the indicated
intervals/points: Interval
d.n.e d.n.e
1. Find the equation of the linear asymptotes of the graph of t , if any. ( 2 points) 2. Give the coordinates of all the relative extrema and points of inflection, if any. ( 2 points) 3. Sketch the graph of the f , emphasizing concavity. Label the intercept points, relative extremum points and points of inflection with their coordinates, and asymptotes with their equations. (5 points
V. Find the antiderivatives and evaluate the definite integral. (4 points each) 1.
3.
2.
VI. Let R be the region bounded by the graphs of the definite integrals giving the
,
, and
as shown below. Set-up
1. Area of R using vertical strips. ( 2 points) 2. Volume of the solid generated when R is revolved about the line with equation a. Using integrals in the variable y . (3 points) b. Using the method of Washers. (3 points) 3. Perimeter of the boundary f=of R using integrals in the variable x . (3 points)
VII.Solve the following problems completely. 1. John and Matt are both walking at 5 feet per second on perpendicular roads, with John walking towards the intersection of the roads and Matt walking away from it. At a certain instant, John is located 60 feet away from the intersection, and 100 feet away from Matt. During this instant, at what rate is the distance between John and Matt changing? Indicate whether the distance is increasing or decreasing. (4 points) 2. As a certain luxury car cruised along a straight road, t exhibited a velocity function of meters per second, where and the positive direction is eastward. The car started from rest. At , the car garnered an acceleration of 8 meters per second squared and was located 2 meters west of a certain marker on the road. a. What is the car’s location at ? ( 2 points) b. Determine the function which describes the position of the car with respect to the marker at each . (4 points) 3. Find such that is minimized. (4 points)
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TOTAL NUMBER OF POINTS : 70
