WW Prob Lib1 Math course-section, semester year

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1 Young-Seon Lee WW Prob Lib Math course-section, semester year WeBWorK assignment due /4/03 at :00 PM..( pt) Give the rational number whose decimal form is: Answer:.( pt) Solve the following inequality: 3 5 Answer: or 3.( pt) Give the equation of the circle whose diameter is determined by points (-,6) and (5,4): Answer: y 4.( pt) Give the equation of the line parallel to y 4 5 that goes through the point 5 : Answer: y 5.( pt) Give the equation of the line perpendicular to y 3 5 that goes through the point 5 : Answer: y 6.( pt) The equation of the line that goes through the points and 3 can be written in the form y m b where m is: and where b is: 7.( pt) Consider the inequality 7 8 The solution of this inequality consists of one or more of the following intervals: A, A B, and B where A B. Find A Find B For each interval, answer YES or NO to whether the interval is included in the solution. A A B B 8.( pt) By completing the square, the epression 4 06 equals A B where A is: and B is:.( pt) For each of the followings angles (in radian measure), find the cos of the angle (your answer cannot contain trig functions, it must be an arithmetic epression or number): 6 4 3

2 Young-Seon Lee WW Prob Lib Math course-section, semester year WeBWorK assignment due /3/03 at :00 PM..( pt) Let F be the function below. If you are having a hard time seeing the picture clearly, click on the picture. It will epand to a larger picture on its own page so that you can inspect it more clearly. 4.( pt) Evaluate the it sin8 0 sin3 5.( pt) Evaluate the it 0 tan 4 Evaluate each of the following epressions. Note: Enter DNE if the it does not eist or is not defined. a) F = b) F = c) F = d) F = e) F = f) F = g) F = h) 3F = i) F 3 =.( pt) Evaluate the it ( pt) Evaluate the it sin3 0 6.( pt) Evaluate the it ( pt) Evaluate the it 8.( pt) Evaluate ( pt) For what value of the constant c is the function f continuous on where f y cy 7 if y 3 cy 7 if y 3 0.( pt) Evaluate the it 6 b b 6 4 b.( pt) Evaluate the it s 8 s 8 s 8.( pt) Evaluate the it t 8 3.( pt) Recall that means: c t 8 t 8 L

3 For all ε 0 there is a δ 0 such that for all satisfying 0 c δ we have that L ε. What if the it does not equal L? Think about what the means in ε δ language. Consider the following phrases:. ε 0. δ c δ 4. L ε 5. but 6. such that for all 7. there is some 8. there is some such that Order these statements so that they form a rigorous assertion that! L c and enter their reference numbers in the appropriate sequence in these boes:

4 Young-Seon Lee WW Prob Lib Math course-section, semester year WeBWorK assignment 3 due //03 at :00 PM..( pt) If find f "#. Find f "# 3..( pt) If find f ". Find f " 3. 4sin cos 3.( pt) If 3tan, find f "#. Find f " 5. 4.( pt) If find f ". Find f " 3. 5.( pt) Let f "# tan sec 6sincos 6.( pt) If 3, find f "#. Find f "#. 7.( pt) If 5 7 4, find f ". Find f "# 3. 8.( pt) If sin, find f "#. Find f "#..( pt) If sin, find f "#. Find f "#. 0.( pt) If tan, find f "$. Find f "# 3..( pt) Let f "#.( pt) Let f "# sin cos ( pt) Let %. Then f "$ 3 is and f "&" 3 is and f "&"&"$ 3 is 4.( pt) Find the 6 th derivative of the function cos. The answer is function 5.( pt) Let f ' 4(

5 Young-Seon Lee WW Prob Lib Math course-section, semester year WeBWorK assignment 4 due /8/03 at :00 PM..( pt) The rate of change of electric charge with respect to time is called current. Suppose that 3 t3 t coulombs of charge flow through a wire in t seconds. (a) Find the current in amperes (coulombs per second) after 3 seconds. (b) When will a 0-ampere fuse in the line blow? a) Current after 3 seconds: amperes. b) A 0-ampere fuse will blow at: seconds..( pt) The radius of a spherical balloon is increasing at the rate of 0.5 inch per second. If the radius is 0 at time t 0, find the rate of change in the volume at time t 3. Rate of change in volume at t 3: inch 3 /second. 3.( pt) Find all points on the graph of y 3 3 where the tangent line has slope. (, ) (, ) Instruction: Enter the points in order of increasing -coordinate. 4.( pt) A space traveller is moving from left to right along the curve y. When she shuts off the engines, she will continue travelling along the tangent line at the point where she is at that time. At what point should she shut off the engines in order to reach the point (4,5)? She should shut off the engine at (, ) 5.( pt) Find all points on the graph of y sincos where the tangent line has horizontal. The tangent of graph is horizontal when + k, where k is an integer. Instruction: There are many ways to epress the answere here. However, WeBWorK is epecting that you choose positive values for both answer boes and the smallest possible value for the first one. 6.( pt) At time t seconds, the center of a bobbing cork is sint centimeters above (or below) water level. What is the velocity of the cork at t 0 )? Velocity at t 0: cm/s. Velocity at t ) : cm/s. Velocity at t : cm/s. 7.( pt) Use implicit differentiation to find the slope of the tangent line to the curve y 5 4 8y at the point 5 4. m 8.( pt) A street light is at the top of a 5 ft. tall pole. A man 6 ft tall walks away from the pole with a speed of 7 feet/sec along a straight path. How fast is the tip of his shadow moving when he is 50 feet from the pole?.( pt) A spherical snowball is melting in such a way that its diameter is decreasing at the rate of 0 4 cm/min. At what rate is the volume of the snowball decreasing when the diameter is 0 cm? (Note the answer is a positive number). 0.( pt) Gravel is being dumped from a conveyor belt at a rate of 30 cubic feet per minute. It forms a pile in the shape of a right circular cone, such that the ratio of the base diameter to the height is always equal to. How fast is the height of the pile increasing when the pile is feet high? Recall that the volume of a right circular cone with height h and radius of the base r is given by V 3 r h.( pt) Use linear approimation to estimate the amount of paint in cubic centimeters needed to apply a coat of paint 0 03 cm thick to a hemispherical dome with a diameter of 70 meters..( pt) Let y 3. Find the change in y, y when 5 and 0 4 Find the differential dy when 5 and d ( pt) Use linear approimation, i.e. the tangent line, to approimate 3 as follows: 3 Let. The equation of the tangent line to at can be written in the form y m b where m is: and where b is:

6 Using this, we find our approimation for 3 is 4.( pt) Einstein s Special Theory of Relativity says that mass m is related to velocity v by the formula m 0 m m 0 v ) c * v, c +. Here, m 0 is the rest mass and c is the velocity of light. Use differentials to determine the percent increase in mass of an object when its velocity increases from 0 5c to 0 54c. Approimate percent increase:

7 Young-Seon Lee WW Prob Lib Math course-section, semester year WeBWorK assignment 5 due 3//03 at :00 PM..( pt) The function is decreasing on the interval (, ). It is increasing on the interval (, ) and the interval (, ). The function has a local maimum at..( pt) For the function f is defined by f On which two intervals is the function increasing? to and to Find the region in which the function is positive: to Where does the function achieve its minimum? 3.( pt) The hands on a clock are of lengths 5 inches (minute hand) and 4 inches (hour hand). How fast is the distance between the tips of the hands changing at 3:00. Rate of change of distance at 3:00 between the tips of the hands: inch(es) per minute. 4.( pt) Identify the critical points and find the maimum value and minimum value of the following function on the given interval. f 3 3, over -. 3) 3. Critical Points:,. Maimum:. Minimum:. Instructions: ) When entering the critical points, please enter them in the order that they appear on the real line. ) If the function has no critical points, enter the string NONE in all answer boes for critical points. 5.( pt) Identify the critical points and find the maimum value and minimum value of the following function on the given interval. 3 f, over Critical Points:,. Maimum:. Minimum:. Instructions: ) When entering the critical points, please enter them in the order that they appear on the real line. ) If the function has no critical points, enter the string NONE in all answer boes for critical points. 6.( pt) What number eceeds its square by the maimum amount? Begin by convincing yourself that this number is on the interval [0,]. Answer:. 7.( pt) A rectangle is to be inscribed in a semicircle of radius r with its base touching that of the semicircle. What are the dimensions of rectangle if its area is to be maimized? Dimensions: r / r. 8.( pt) Answer the following questions for the function 5 defined on the interval A. is concave down on the region to B. is concave up on the region to C. The inflection point for this function is at D. The minimum for this function occurs at E. The maimum for this function occurs at.( pt) Answer the following questions for the function 3 36 defined on the interval Enter points, such as inflection points in ascending order, i.e. smallest values first. Enter intervals in ascending order also. A. The function has vertical asympototes at and B. is concave up on the region to and to C. The inflection points for this function are, and 0.( pt) Consider the function 3% 7 3% 3. For this function there are two important intervals: A and A where the function is not defined at A. Find A

8 For each of the following intervals, tell whether is increasing (type in INC) or decreasing (type in DEC). A : A Note that this function has no inflection points, but we can still consider its concavity. For each of the following intervals, tell whether is concave up (type in CU) or concave down (type in CD). A : A.( pt) A fence 6 feet tall runs parallel to a tall building at a distance of 5 feet from the building. What is the length of the shortest ladder that will reach from the ground over the fence to the wall of the building?.( pt) Let Q 0 5 and R 0 be given points in the plane. We want to find the point P 0 on the -ais such that the sum of distances PQ PR is as small as possible. (Before proceeding with this problem, draw a picture!) To solve this problem, we need to minimize the following function of : over the closed interval - a b where a and b. We find that has only one critical number in the interval at where has value Since this is smaller than the values of at the two endpoints, we conclude that this is the minimal sum of distances.

9 Young-Seon Lee WW Prob Lib Math course-section, semester year WeBWorK assignment 6 due 4/5/03 at :00 PM. s s.( pt) Evaluate the integral: ds. s Answer: + C..( pt) Evaluate the indefinite integral: 3y y 5 dy. Answer: + C. 3.( pt) Find: sin d. Answer: + C. 4.( pt) A ball is shot at an angle of 45 degrees into the air with initial velocity of 47 ft/sec. Assuming no air resistance, how high does it go? How far away does it land? Hint: The acceleration due to gravity is 3 ft per second squared. 5.( pt) Consider the function f t 7sec t 3 6t 3. Let F t be the antiderivative of f t with F 0 0. Then F 4 6.( pt) Consider the function whose second derivative is f "&"# 7sin. If f 0 4 and f "# 0, what is f? 7.( pt) Consider the function Enter an antiderivative of 8.( pt) A particle is moving with acceleration a t 36t 0. Its position at time t 0 is s 0 3 and its velocity at time t 0 is v 0 4. What is its position at time t 6?.( pt) A stone is dropped from the edge of a roof, and hits the ground with a velocity of 5 feet per second. How high (in feet) is the roof? 0.( pt) Consider the differential equation: dy 54 d y. a) Find the general solution to the above differential equation. (Instruction: Call your integration constant C.) Answer: y. b) Find the particular solution of the above differential equation that satisfies the condition y 4 at. Answer: y..( pt) Consider the differential equation: du u 3 t 3 t. dt a) Find the general solution to the above differential equation. (Instruction: Write the answer in a form such that its numerator is and its integration constant is C rename your constant if necessary.) Answer: u. b) Find the particular solution of the above differential equation that satisfies the condition u 4 at t 0. Answer: u..( pt) An object is moving along a coordinate line subject to acceleration a (in centimeters per second per second) as follows a t 4 with initial velocity v 0 0 (in centimeters per second) and directed distance s 0 0 (in centimeters). Find both the velocity v and the directed distance s after seconds. Velocity after seconds: centimeter(s) per second. Directed distance after seconds: centimeter(s). 3.( pt) Find 7 k k k6 3 k =. 4.( pt) Find 6 k67 k sin k) =. 5.( pt) Find the value of the following collapsing sum: 0 k6 k k =. 6.( pt) Use the Special Sum Formulas (see Section 5.3 of Varberg, Purcell and Rigdon) to find: 0 i6 i 8 4i 3 =.

10 * Young-Seon Lee WW Prob Lib Math course-section, semester year WeBWorK assignment 7 due 4/6/03 at :00 PM..( pt) 0 : 0 0 where a= and b= a b.( pt) Consider the function 4. In this problem you will calculate 0 4 d by using the definition b n d f a n ; i < i6 The summation inside the brackets is R n which is the Riemann sum where the sample points are chosen to be the right-hand endpoints of each sub-interval. Calculate R n for 4 on the interval - 0 and write your answer as a function of n without any summation signs. You will need the summation formulas in Section 5.3 of your tetbook. R n n R n 3.( pt) If f() = then f "# f "# 4.( pt) Given 3 3 t3 dt 0 t 4 cos t At what value of does the local ma of occur? dt 5.( pt) Let G t dt Find G"$ 6.( pt) View the following it as a definite integral and then evaluate that integral by the Second Fundamental Theorem of Calculus. n n n ; i n i n + <= i6 The above it is equal to. 7.( pt) Use the method of substitution to find the following definite integral:, cosθcos sinθ dθ, Answer:. 8.( pt) Sketch the region enclosed by the given curves. Decide whether to integrate with respect to or y. Then find the area of the region. y 0 y 0.( pt) Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified ais. y 6 y ; about y 0

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