Math 102 Calculus 1 Instructional Materials (Notes and Exercises) PDF

Department of Mathematics and Statistics College of Science Polytechnic University of the Philippines Math 102 - Calculus 1 Instructional Materials (Notes and Exercises) Unit 3 Prepared by: John Patrick B. Sta. Maria [email protected] Lesson 3.1...

Department of Mathematics and Statistics College of Science Polytechnic University of the Philippines Math 102 - Calculus 1 Instructional Materials (Notes and Exercises) Unit 3 Prepared by: John Patrick B. Sta. Maria [email protected] Lesson 3.1 Math 102 Lesson 3.1 Related Rates Problems NOTES: DEFINITION. Let x and y be variables which are both differentiable functions of the variable t, i.e., there exist differentiable functions f and g such that x = f (t) and y = g(t). If x and y may be related by the relation F (x, y) = 0, then the derivatives dx dt and dy dt are said to be related rates. REMARKS: 1. The variable t usually represents the time variable and that x and y are related variables which both change (smoothly) with respect to time. 2. There could be more than two time-dependent variables involved in a situation. Common Related Rates Problems 1. A point moving along a curve: The x and y coordinates changes with respect to time. 2. Growing ripples in the pond caused by dropping a stone: Both the area and radius of the ripples are growing with respect to time. 3. A sliding ladder leaning against a wall: As the distance of the top of the ladder from the ground is decreasing, the distance of the foot from the wall is increasing. 4. Water pouring into a container with specific shape: The cross section of the water changes as the water level increases. 5. Two objects moving in different directions: Their distance changes with respect to time de- pendent on how fast/slow they are moving. 6. A human moving away or towards a light source: This causes the length of the shadow to vary over time. Suggested Steps in Solving Related Rates Problems 1. If possible, draw a diagram of the problem which represents the situation for any t > 0. 2. Name variables which represent quantities that change with respect to time. Label the diagram. 3. Identify the quantities which have constant values (that do not depend on t) and label the diagram. 4. Identify what is asked in the problem. 5. Find an equation that relates all the variables that change with respect to t. 6. Differentiate both sides of the equation obtained in the previous step with respect to t im- plicitly and substitute the values that are valid at a particular time. 7. Solve what is asked and write a conclusion that directly answers the question in the problem. Include correct units of measurement. Page 2 of 17 jpbstamaria Lesson 3.1 Math 102 EXERCISES: 1. A point P is moving along the parabola y = x2 + 2x − 3 in such a way that the x coordinate changes at a constant rate of 2 units per second. At what rate is the y-coordinate of P changing at the instant when x = 1? 2. A point Q moves in the circumference of an ellipse whose equation is x2 + 4y 2 = 8. At what rate is the x-coordinate of Q changing at the instant when it is at the point (2, 1) given that the y-coordinate is changing at 0.5 unit per second. 3. A cylindrical tank with radius 5 m is being filled with water at a rate of 3 m3 /min. How fast is the water level increasing? 4. Water is pouring into an inverted cone at a rate of 8 cubic feet per minute. If the height of the cone is 12 feet and the radius of its base is 6 feet, how fast is the water level rising when the water is 4 feet deep? 5. If a snowball melts so that its surface area decreases at a rate if 1 cm3 /min, find the rate at which the diameter decreases when the diameter is 10 cm. 6. A street light is mounted at the top of a 15-foot-tall pole. A man 6 feet tall walks away from the pole at a speed of 5 ft/sec along a straight path. How fast is the length of shadow changing? 7. A ladder 25 feet long is leaning against the wall of a building. An oil spill causes the foot of the ladder to slide away from the wall at a rate of 2 feet per second. (a) How fast is the top of the ladder moving down the wall when its foot is exactly 7 feet from the wall? (b) How fast is the angle formed by the ladder and the ground changing the moment when the top of the ladder is 24 feet from the ground? 8. A ladder 15 ft long rests against a vertical wall. Its top slides down the wall while its bottom moves away along the level ground at a speed of 2 ft/s. How fast is the angle between the top of the ladder and the wall changing when the angle is π/3 radians? 9. A ladder 12 meters long leans against a wall. The foot of the ladder is pulled away from the wall at the rate 21 m/min. At what rate is the top of the ladder falling when the foot of the ladder is 4 meters from the wall? 10. A lighthouse is located 3 km west of a port. A boat leaves the port at a constant speed of 5 km per minute heading north along a straight line. At what rate is the distance of the boat from the lighthouse changing the moment when the boat is 4 km from the port? 11. A rocket R is launched vertically and its tracked from a radar station S which is 4 miles away from the launch site at the same height above sea level. At a certain instant after launch, R is 5 miles Page 3 of 17 jpbstamaria Lesson 3.1 Math 102 away from S and the distance from R to S is increasing at a rate of 3600 miles per hour. Compute the vertical speed v of the rocket at this instant. 12. A boat is pulled into a dock by means of a rope attached to a pulley on the dock, see figure below. The rope is attached to the bow of the boat at a point 1 m below the pulley. If the rope is pulled through the pulley at a rate of 1 m/sec, at what rate will the boat be approaching the dock when 10 m of rope is out. Pulley Boat Dock 13. A person (A) situated at the edge of the river observes the passage of a speed boat going down- stream. The boat travels exactly through the middle of the river, which is 10 m wide. When the boat is at θ = 60◦ (see figure) the observer measures the rate of change of the angle θ to be 2 radians/second. What is the speed of the boat at that instant? boat θ A 14. An airplane flying horizontally at an altitude of y = 3 km and at a speed of 480 km/h passes directly above an observer on the ground. How fast is the distance s from the observer to the airplane increasing 30 seconds later? 15. An airplane flying horizontally at a constant height of 1000 m above a fixed radar station. At a certain instant the angle of elevation θ at the station is π/4 radians and decreasing at a rate of 0.1 rad/sec. What is the speed of the aircraft at this moment. 16. A kite is rising vertically at a constant speed of 2 m/s from a location at ground level which is 8 m away from the person handling the string of the kite. kite z y θ 8m Page 4 of 17 jpbstamaria Lesson 3.1 Math 102 (a) Let z be the distance from the kite to the person. Find the rate of change of z with respect to time t when z = 10. (b) Let θ be the angle the string makes with the horizontal. Find the rate of change of θ with respect to time t when the kite is 6 m above ground. 17. A balloon is rising at a constant speed 4 m/sec. A boy is cycling along a straight road at a speed of 8m/sec. When he passes under the balloon, it is 36 meters above him. How fast is the distance between the boy and balloon increasing 3 seconds later? 18. A helicopter takes off from a point 80 m away from an observer located on the ground, and rises vertically at 2 m/s. At what rate is elevation angle of the observer’s line of sight to the helicopter changing when the helicopter is 60 m above the ground. 19. An oil slick on a lake is surrounded by a floating circular containment boom. As the boom is pulled in, the circular containment boom. As the boom is pulled in, the circular containment area shrinks (all the while maintaining the shape of a circle.) If the boom is pulled in at the rate of 5 m/min, at what rate is the containment area shrinking when it has a diameter of 100 m? 20. Consider a cube of variable size. (The edge length is increasing.) Assume that the volume of the cube is increasing at the rate of 10 cm3 /minute. How fast is the surface area increasing when the edge length is 8 cm? 21. The height of a rectangular box is increasing at a rate of 2 meters per second while the volume is decreasing at a rate of 5 cubic meters per second. If the base of the box is a square, at what rate is one of the sides of the base decreasing, at the moment when the base area is 64 square meters and the height is 8 meters? 22. Sand is pouring out of a tube at 1 cubic meter per second. It forms a pile which has the shape of a cone. The height of the cone is equal to the radius of the circle at its base. How fast is the sandpile rising when it is 2 meters high? 23. A water tank is in the shape of a cone with vertical axis and vertex downward. The tank has radius 3 m and is 5 m high. At first the tank is full of water, but at time t = 0 (in seconds), a small hole at the vertex is opened and the water begins to drain. When the height of water in the tank has dropped to 3 m, the water is flowing out at 2 m3 /s. At what rate, in meters per second, is the water level dropping then? 24. A boy starts walking north at a speed of 1.5 m/s, and a girl starts walking west at the same point P at the same time at a speed of 2 m/s. At what rate is the distance between the boy and the girl increasing 6 seconds later? 25. At noon of a certain day, the ship A is 60 miles due north of the ship B. If the ship A sails east at speed of 15 miles per hour and B sails north at speed of 12.25 miles per hour, determine how rapidly the distance between them is changing 4 hours later? Page 5 of 17 jpbstamaria Lesson 3.1 Math 102 26. A lighthouse is located on a small island three km off-shore from the nearest point P on a straight shoreline. Its light makes four revolutions per minute. How fast is the light beam moving along the shoreline when it is shining on a point one km along the shoreline from P ? 27. A police car, approaching right-angled intersection from the north, is chasing a speeding SUV that has turned the corner and is now moving straight east. When the police car is 0.6 km north of intersection and the SUV is 0.8 km east of intersection, the police determine with radar that the distance between them and the SUV is increasing at 20 km/hr. If the police car is moving at 60 km/hr at the instant of measurement, what is the speed of the SUV? Page 6 of 17 jpbstamaria Lesson 3.2 Math 102 Lesson 3.2 Extrema of Functions and Optimization Problems NOTES: DEFINITION. A function f is said to have a 1. relative maximum at x = a if there exists an open interval I containing a such that f (x) ≤ f (a) for all x ∈ I. 2. relative minimum at x = a if there exists an open interval I containing a such that f (x) ≥ f (a) for all x ∈ I. 3. relative extremum at x = a if f has a relative minimum or a relative maximum at x = a. f a1 a2 a3 a4 f has rel. max. at x = a1 and x = a3. f has rel. min. at x = a2 and x = a4. LEMMA. Suppose that lim F (x) = L exists. x→a 1. If L > 0, then there exists an open interval I containing a such that f (x) > 0 for all x ∈ I, x 6= a. 2. If L < 0, then there exists an open interval I containing a such that f (x) < 0 for all x ∈ I, x 6= a. FERMAT’S THEOREM. Suppose that a function f has a relative extremum at x = a. If f 0 (a) exists, then f 0 (a) = 0. This means that at points where f attains a relative extremum, either f 0 (a) does not exist or f 0 (a) = 0 (has a horizontal tangent line at x = a). DEFINITION. A point a in the domain of f is called a critical number (of f ) if either f 0 (a) does not exist or f 0 (a) = 0. The ordered pair (a, f (a)) is called a critical point. COROLLARY. If f has a relative extremum at x = a, then a is a critical number of f. REMARK: The converse is false. The critical points where f doesn’t have a relative extremum is called a stationary point. Page 7 of 17 jpbstamaria Lesson 3.2 Math 102 DEFINITION. Let I be the domain of f and let a ∈ I. A function f achieves an 1. absolute maximum at x = a if f (x) ≤ f (a) for all x ∈ I. 2. absolute minimum at x = a if f (x) ≥ f (a) for all x ∈ I. 3. absolute extremum at x = a if f has an absolute maximum or absolute minimum at x = a. EXTREME VALUE THEOREM (EVT). If f is continuous on [a, b], then f achieves both an absolute maximum and absolute minimum values at certain points in [a, b]. In other words, there exist r, s ∈ [a, b] such that f (r) ≤ f (x) ≤ f (s), for all x ∈ [a, b]. COROLLARY. Let f be continuous on [a, b]. Suppose that the set of all critical numbers of f in (a, b) is {c1 , c2 ,... , cn }. Let S = {f (a), f (c1 ), f (c2 ),... , f (cn ), f (b)}. The largest number in S is the absolute maximum of f on [a, b] and the least number in S is the absolute minimum of f on [a, b]. THEOREM. Suppose that a function f is continuous on an interval I containing a and that a is the only number in I, where f has a relative extremum. 1. If f has a relative maximum at x = a, then f has an absolute maximum at x = a on I. 2. If f has a relative minimum at x = a, then f has an absolute minimum at x = a on I. DEFINITION. A 1-dimensional optimization problem involves maximizing or minimizing an objective function y = f (x) subject to a constraint on the values of x. Suggested Steps in Solving Optimization Problems. 1. Draw a diagram which best represent the situation in the problem. 2. Assign variables to the quantities involved. 3. Identify the objective function and determine whether the problem is a maximization or minimization problem. 4. Determine the constraints from the physical restrictions presented in the problem. 5. Use appropriate theorems involving extreme values to solve the problem and make a conclu- sion that directly answers the problem. Use appropriate units of measurements. Page 8 of 17 jpbstamaria Lesson 3.2 Math 102 EXERCISES: I. In each of the following, find the absolute maximum and absolute minimum values of f (x) in the indicated interval and specify at what values of x at which these extrema are achieved. √ (a) f (x) = 3x2 − 9x on [−1, 2] (j) f (x) = x 4 − x on [0, 4] (b) f (x) = x3 − 3x2 + 1 in [− 21 , 4] (k) f (x) = 1 + |x − 4| on [0, 6] (c) f (x) = x3 − 12x − 5 on [−4, 6] (l) f (x) = (x − 1)2/3 on [−7, 2] (d) f (x) = 3x4 − 4x3 on [−1, 2] (m) f (x) = 2 sin x − cos 2x on [0, 2π] (e) f (x) = x4 − 4x2 on [−3, 3] (n) f (x) = x − tan x on [− π4 , π4 ] (f) f (x) = x4 − 4x on R (o) f (x) = 2 sec x + tan x on [0, 2π] 1 sin x (g) f (x) = on [−2, 1] (p) f (x) = √ on [0, 2π] 1 + x2 2 + cos x 1 (q) f (x) = sin(cos x) on [0, 2π] (h) f (x) = on (−2, 1) 1 + x2 (r) f (x) = x ln x on (0, ∞) 16 √ (i) f (x) = x + on (1, ∞) (s) f (x) = ln(1 + 3x2 ) + 2 tan−1 ( 3x) on R x II. Prove that the function  x2 − 4, if x < 3 f (x) = 8 − x, if x ≥ 3 is continuous on [0, 5] and determine its absolute extrema on [0, 5]. III. Let a, b > 0. At what value of x does the function f (x) = xa (1−x)b achieves its absolute maximum value on [a, b]? IV. Consider the function f (x) = x1/x , for x > 0. (a) Show that f (x) has an absolute maximum value on (0, ∞) at x = e. (b) Use (a) to prove that eπ > π e. V. Solve each of the following optimization problems. (a) Find two nonnegative numbers such that their sum is 20 and that their product is as large as possible. (b) What is the area of the largest rectangle which can be inscribed in a circle of radius 1 unit? (c) Find the area of the rectangle with maximum perimeter and with sides parallel to the co- ordinate axes that can be inscribed in the region enclosed by the graphs of y = x2 and y = 4. (d) Find the dimensions of the rectangle of largest area that has its base on the x-axis and its other two vertices above the x-axis and lying on the parabola y = 12 − x2. Page 9 of 17 jpbstamaria Lesson 3.2 Math 102 (e) A rectangular box is to be made from a piece of cardboard 24 inches long and 9 inches wide by cutting out identical squares from the four corners and turning up the sides. Find the volume of the largest rectangular box that can be formed. (f) You are in a rowboat 2 miles from a straight shoreline and you notice a smoke billowing from your house, which is on the shoreline and 6 miles from the point on the shoreline nearest to you. If you can row at 6 mph and run at 10 mph, how should you proceed in order to get to his house in the least amount of time? (g) Find the equation of the tangent line to the curve y = x3 −3x2 +5x that has the least possible slope. (h) A four feet wire is to be cut at some point P forming two smaller wires. A circle and a square is to be constructed from the two smaller wires. Where should the point P be located so that the combined area of the circle and the square is as large as possible? (i) A straight piece of wire of 28 cm is cut into two pieces. One piece is bent into a square. The other piece is bent into a rectangle with aspect ratio 1 : 3 What are dimensions, in centimeters, of the square and the rectangle such that the sum of their areas is minimized? (j) With a straight piece of wire 4 m long, you are to create an equilateral triangle and a square, or either one only. Suppose a piece of length x meters is bent into triangle and the reminder is bent into a square. Find the value of x which maximizes the total area of both triangle and square. (k) Find a positive number such that its sum with its reciprocal is the least. (l) A rectangular page is to contain 30 squared inches of print. The margins on each side is 1 in. Find the dimensions of the page such that the least amount of paper is used. (m) Each rectangular page of a book must contain 30 cm2 of printed text, and each page must have 2 cm margins at top and bottom, and 1 cm margin at each side. What is the minimum possible area of such a page? (n) Find the point on the curve x + y 2 = 0 that is closest to the point (0, −3). (o) A farmer has 400 feet of fencing with which to build a rectangular pen. He will use part of an existing straight wall 100 feet long as part of one side of the perimeter of the pen. What is the maximum area that can be enclosed? (p) Maya is 2 km offshore in a boat and wishes to reach a coastal village which is 6 km down a straight shoreline from the point on the shore nearest to the boat. She can row at 2 km/hr and run at 5 km/hr. Where should she land her boat to reach the village in the least amount of time? (q) A rectangular box has a square base with edge length x of at least 1 unit. The total surface area of its six sides is 150 square units. Find the dimensions of the box with the greatest possible volume. What is this greatest possible volume? Page 10 of 17 jpbstamaria Lesson 3.2 Math 102 (r) An open-top box is to have a square base and a volume of 10 m3. The cost per square meter of material is PHP 5.00 for the bottom and PHP 2.00 for the four sides. Find the dimensions of the box so that the cost of materials is minimized. What is this minimum cost? (s) An open-top box is to have a square base and a volume of 13,500 cm3. Find the dimensions of the box that minimize the amount of material used. (t) Find the dimension of the right circular cylinder of maximum volume that can be inscribed in a right circular cone of radius R and height H. (u) A hollow plastic cylinder with a circular base and open top is to be made and 10 m2 plastic is available. Find the dimensions of the cylinder that give the maximum volume and find the value of the maximum volume. (v) An open-topped cylindrical pot is to have volume 250 cm3.The material for the bottom of the pot costs PHP 4.00 per cm2 ; that for its curved side costs PHP 2.00 per cm2. What dimensions will minimize the total cost of this pot? (w) Show that the volume of the largest cone that can be inscribed inside a sphere of radius r is 32πr3. 81 (x) The sound level measured in watts per square meter, varies in direct proportion to the power of the source and inversely as the square of the distance from the source, so that is given by y = kP x−2 , where y is the sound level, P is the source power, x is the distance form the source, and k is a positive constant. Two beach parties, 100 meters apart, are playing loud music on their portable stereos. The second party’s stereo has 64 times as much power as the first. The music approximates the white noise, so the power from the two sources arriving at a point between them adds, without any concern about whether the sources are in or out of phase. To what point on the line segment between the two parties should I go, if I wish to enjoy as much quiet as possible? Demonstrate that you have found an absolute minimum, not just a relative minimum. Page 11 of 17 jpbstamaria Lesson 3.3 Math 102 Lesson 3.3 The Mean Value Theorem and the Derivative Tests NOTES: ROLLE’S THEOREM. Let f be a function satisfying the following conditions: f 1. f is continuous on [a, b]; 2. f is differentiable on (a, b); and 3. f (a) = f (b). a c b Then there exists c ∈ (a, b) such that f 0 (c) = 0. MEAN VALUE THEOREM. Let f be a function sat- f isfying the following conditions: 1. f is continuous on [a, b]; 2. f is differentiable on (a, b). Then there exists c ∈ (a, b) such that f (b) − f (a) = f 0 (c)(b − a). a c b CONSEQUENCES OF THE MEAN VALUE THEOREM 1. If f 0 (x) = 0 for all x ∈ I, where I is an open interval, then f is constant on I. 2. If f 0 (x) > 0 for all x ∈ I, where I is an open interval, then f is strictly increasing on I. 3. If f 0 (x) < 0 for all x ∈ I, where I is an open interval, then f is strictly decreasing on I. THEOREM. (THE FIRST DERIVATIVE TEST) Let c be a critical number of f such that f is continuous in an open interval I containing c. If f is differentiable at every point in I \ {c}, then f (c) can be classified as follows. 1. If f 0 (x) > 0, for all x ∈ I, x < c and f 0 (x) < 0, for all x ∈ I, x > c, then f has a relative maximum f (c) attained at c. 2. If f 0 (x) < 0, for all x ∈ I, x < c and f 0 (x) > 0, for all x ∈ I, x > c, then f has a relative minimum f (c) attained at c. 3. If f 0 (x) > 0, for all x ∈ I, x 6= c, then f doesn’t have a relative extremum at c. 4. If f 0 (x) < 0, for all x ∈ I, x 6= c, then f doesn’t have a relative extremum at c. NOTES ON FDT: 1. If f 0 changes from + to − across x = c, then f has a relative maximum at x = c. 2. If f 0 changes from − to + across x = c, then f has a relative minimum at x = c. Page 12 of 17 jpbstamaria Lesson 3.3 Math 102 DEFINITION. (CONCAVITY) Let f be differentiable on an open interval I. We say that the graph of f is 1. concave upward on I if f 0 is increasing on I. 2. concave downward on I if f 0 is decreasing on I. The point (c, f (c)) is called a point of inflection of f if f is continuous at x = c and the concavity of f changes across x = c. LEMMA. Let f be a function which is twice-differentiable on an open interval I. 1. If f 00 (x) > 0 on I, then the graph of f is concave upward on I. 2. If f 00 (x) < 0 on I, then the graph of f is concave downward on I. 3. If (c, f (c)) is a point of inflection of the graph of f , where c ∈ I, then f 00 (c) = 0. THEOREM (SECOND DERIVATIVE TEST). Suppose that f be twice differentiable on I, where I is an open interval containing c and that f 0 (c) = 0. 1. If f 00 (c) > 0, then f has a relative minimum at x = c. 2. If f 00 (c) < 0, then f has a relative maximum at x = c. GUIDELINES FOR SKETCHING GRAPHS 1. Domain of f. Find the domain of f. 2. Continuity of f. Determine whether f is continuous on its domain and if not, classify the discontinuities. 3. Intercepts. Identify the x-intercepts by solving f (x) = 0 and the y-intercept by calculating f (0). 4. Symmetry. If f is an even function [f (−x) = f (x)], the graph is symmetric with respect to the y-axis. If f is an odd function [f (−x) = −f (x)], the graph is symmetric with respect to the origin. 5. Asymptotes. Determine whether f has vertical asymptotes, horizontal asymptotes, slant asymptotes, or non-linear asymptotes. 6. Critical Numbers and Local Extrema. Find f 0 (x) and the critical numbers. Use FDT to identify relative extrema. Identify whether there are cusps or sharp turns. 7. Concavity and Points of Inflection. Find f 00 (x) and use the SDT, when necessary. Determine the intervals where f is concave upward and when f is concave downward. Also locate points of inflection if there is any. Page 13 of 17 jpbstamaria Lesson 3.3 Math 102 CAUCHY’S MEAN VALUE THEOREM. Let f and g be functions satisfying the following conditions: 1. f and g are continuous on [a, b]; and 2. f and g are differentiable on (a, b). Then there exists c ∈ (a, b) such that f (b) − f (a) f 0 (c) = 0 , g(b) − g(a) g (c) provided that the equation above makes sense. THEOREM. (L’HÔPITAL’S RULE). Let I be an open interval containing a. Suppose that lim f (x) and lim g(x) are both zero or both ∞. If f and g are both differentiable on I and that x→a x→a 0 f 0 (x) g (x) 6= 0 for all x ∈ I \ {a}, and limx→a g 0 (x) exists, then f (x) f 0 (x) lim = lim 0. x→a g(x) x→a g (x) Page 14 of 17 jpbstamaria Lesson 3.3 Math 102 EXERCISES: I. Determine if f satisfies the hypotheses of the Rolle’s Theorem on the indicated interval. If so, determine the value of c as in the conclusion of the Rolle’s Theorem. (a) f (x) = 3x2 − 12x + 11 on [0, 4]. (b) f (x) = x4 + 4x2 + 1 on [−3, 3]. (c) f (x) = cos 2x + 2 cos x on [0, 2π]. (d) f (x) = 1 − |x| on [−1, 1]. (e) f (x) = ex + e−x on [−2, 2]. II. Determine if the mean value theorem applies to the following functions on the indicated intervals. If so, determine the value(s) of c as in the conclusion of the mean value theorem. (a) f (x) = x2 + 2x − 1 on [0, 1]. 4 (b) f (x) = on [5/2, 4]. 9 − 2x (c) f (x) = 3(x − 4)2/3 on [−4, 5]. √ (d) f (x) = 1 + cos x on [−π/2, π/2]. (e) f (x) = ex cosh x on [0, ln 2]. III. Do as instructed. (a) Two cars start a race at the same time and finish in a tie. Prove that at some time during the race, they have the same speed. (b) Apply the Rolle’s theorem to f (x) = x4 − 2x2 + 4x on [−2, 0] and conclude that x3 − x + 1 = 0 has a solution on [−2, 0]. (c) Let n > 2 be an odd positive integer. Prove that the equation xn + nx + 1 = 0 has exactly one real root. (d) Let f and g be functions such that f 0 (x) = g 0 (x) for all x ∈ I, where I is an open interval. Prove that there exists a constant C such that f (x) = g(x) + C for all x ∈ I. (e) Consider the function f (x) = tan−1 x. i. Prove that for any a, b where a < b, we have b−a b−a 2 < tan−1 b − tan−1 a <. 1+b 1 + a2 π 3 4 π 1 ii. Show that + < tan−1 < +. 4 25 3 4 6   a b b (f) Let a < b. Use the mean value theorem to prove that 1 − < ln < − 1. b a a Page 15 of 17 jpbstamaria Lesson 3.3 Math 102 (g) Use the mean value theorem to prove that √   π 3 −1 3 π 1 + < sin < +. 6 15 5 6 8 r 1−x ln(1 + x) (h) Prove that for 0 < x < 1, we have < < 1. 1+x sin−1 x (i) Let f be twice differentiable on (a, b) and suppose that f 0 (a) = f 0 (b) = 0. Prove that there 4 exists c ∈ (a, b) such that |f 00 (c)| ≥ [f (b) − f (a)]. (b − a)2 (j) Prove that between any two real roots of ex sin x = 1, there exists a real root of ex cos x = −1. IV. Given that x2 − 4x 0 4(3x − 4) 24(x − 4) f (x) = 2 , f (x) = 2 , and f 00 (x) = −. (x + 4) (x + 4) (x + 4)4 (a) Find the x and y-intercepts of the graph of f. (b) Determine the asymptotes of the graph of f. (c) Determine the open intervals where the function is strictly increasing or strictly decreasing. (d) Identify the relative extrema of f , if there’s any. (e) Determine the open intervals where the graph of the function is concave upward or concave downward. (f) Identify the point/s of inflection of the graph of f , if there’s any. (g) Sketch the graph of f. V. Graph the following functions by considering the guidelines for sketching graphs. 5
(a) f (x) = 3x4 − 8x3 + 10 (f) f (x) = x2/3 2 −x (b) f (x) = 3x6 − 6x3 (g) f (x) = 3 sin x − sin3 x on [0, 2π] x2 − 1 (c) f (x) = (h) f (x) = 6x + sin(3x) on [0, 2π] x x3 (i) f (x) = x2 e−x (d) f (x) = 2 x −1 1 (e) f (x) = 4x1/3 + x4/3 (j) f (x) = (1 + ex )2 VI. Evaluate the following limits. x − sin x (d) lim x3 e−2x (a) lim x→∞ x→0 x3 e2x − 2ex + 1 (e) lim+ x3 ln x (b) lim x→0 x→0 cos(3x) − 2 cos(2x) + cos x 3x − 2x  πx  (c) lim+ (x2 − 1) tan (f) lim x→1 2 x→0 x Page 16 of 17 jpbstamaria Lesson 3.3 Math 102 tan−1 x − sin−1 x  2x 3 (g) lim 1 − (l) lim x→∞ x x→0 x(1 − cos x)   (h) lim (1 + 2x)1/3x x+3 x→∞ (m) lim x ln   x→∞ x−3 1 1/x2 (i) lim − csc x  sin x x→0 x (n) lim x→0 x (j) lim xsin x x→0 (o) lim (x + ex + e2x )1/x   x→∞ 1 2 (k) lim − cot x (p) lim+ (sin x)1/ ln x x→0 x2 x→0 Page 17 of 17 jpbstamaria