Study Guide For Final Exam Fall 2024 PDF

Study Guide for Final Exam (Version: 8:00 PM, Thursday, November 28) New materials after Exam 3 Remark: There will be 5 questions regarding the new mate- rials after Exam 3 in the Final Exam. Look also at the “NOTE” for the subject matter 14 about computing the total distance. 1....

Study Guide for Final Exam (Version: 8:00 PM, Thursday, November 28) New materials after Exam 3 Remark: There will be 5 questions regarding the new mate- rials after Exam 3 in the Final Exam. Look also at the “NOTE” for the subject matter 14 about computing the total distance. 1. You are supposed to know how to compute the integral using the Riemann Sum. Conversely, you should know how to compute the limit in the form of Riemann Sum using the integration and Fundamental Theorem of Calculus. You are also supposed to know how to compute the integral knowing its geometrical meaning. Example Problems 1.1. Z 1 √ (i) Write down the formula for approximating the integration 1 − x2 dx as the Riemann Sum dividing the interval [0, 1] into n 0 equal subintervals and using the right end points. (ii) Compute the integral using the geometric interpretation. 1.2. Evaluate the following definite integrals Z 5p Z 3+2√2 p (i) 16 − (x − 3)2 dx (ii) 16 − (x − 3)2 dx. 3 −1 √ Z 1 √  Z 3 √ (iii) 1 + 1 − x2 dx (iv) {1 + 4 − x2 } dx −1 0 Z 5 1.3. Evaluate the following definite integral |3x − 1| dx. 0 1.4. Computethe following limits:  Pn k (i) limn→∞ k=1 2 n + k2 HINT: Use the equation Pn k Pn k/n n Pn k/n 1 k=1 2 2 = k=1 2 · 2 = k=1 2 · n +k Pn 1 +∗(k/n) n 1 + (k/n) n = k=1 f (x k )∆x where  x  f (x) =   1 + x2 ∗ xk = k/n  ∆x = 1 = b − a.   n n 1 2 Now n X Z 1 lim f (x∗k )∆x = f (x) dx. n→∞ 0 k=1   Pn 1 (ii) limn→∞ k=1 n + 4k HINT: Use the equation Pn 1 Pn 1 1 k=1 = k=1 · n + 4k Pn 1 +∗4 (k/n) n = k=1 f (xk )∆x where 1    f (x) = 1 + 4x   x∗k = k/n   1 b−a  ∆x =  =. n n Now n Z 1 X ∗ lim f (xk )∆x = f (x) dx. n→∞ 0 i=1   Pn n (iii) limn→∞ k=1 n + k2 2 HINT: Use the equation Pn n Pn 1 n Pn 1 1 k=1 = k=1 2 · 2 = k=1 2 · n2 +k 2 Pn 1 +∗(k/n) n 1 + (k/n) n = k=1 f (xk )∆x where 1   f (x) = 1 + x2    x∗k = k/n   1 b−a  ∆x =  = n n Now n Z 1 X ∗ lim f (xk )∆x = f (x) dx. n→∞ 0 k=1  3   Pn 1 (iv) limn→∞ k=1 √ 4n2 − k 2 HINT: Use the equation Pn 1 Pn 1 1 1 1/2 = nk=1 s P k=1 √ = k=1 q · · 4n2 − k 2 2n 2 n 1 − (k/2n)2   1/2 1− k· n Pn ∗ = k=1 f (xk )∆x where 1    f (x) = √ 1 − x2    1/2  ∗ x k = k ·  n 1/2 b−a     ∆x =  =. n n Now n 1/2 1/2 1 X Z Z lim f (x∗k )∆x = f (x) dx = √ dx. n→∞ k=1 0 0 1 − x2 Alternatively, one may use the equation Pn 1 Pn 1 1 P k=1 √ = k=1 q · = nk=1 f (x∗k )∆x 4n2 − k 2 n 4 − (k/n)2 where 1    f (x) = √ 4 − x2   x∗k = k/n  ∆x = 1 = b − a.    n n Now n Z 1 Z 1 X ∗ 1 lim f (xk )∆x = f (x) dx = √ dx. n→∞ k=1 0 0 4 − x2 We observe Z 1 Z 1 Z 1/2 1 1/2 1 √ dx = p dx = √ du, 4−x 2 1 − (x/2) 2 1 − u2 0 0 0 where the last equality is obtained  by the substitution  u = x/2. Pn 1 (v) (Extra Practice) limn→∞ k=1 √ 2 n − k2 4 2. You are supposed to understand the meaning of the Fundamental Theorem of Calculus, and use it to compute the derivative of a function given in the form of an integration. (This includes the requirement that you are supposed to know how to take an anti-derivative of a function.) You are also supposed to understand the special features of the integration involving even/odd functions. Example Problems Z x2 2.1. Set F (x) = sin(πt) dt. Compute F 0 (1/2). x 2.2. Compute the following. ! Z x4 Z 3x 2  d d t −1 (i) sec(t)dt (ii) 2 dt dx 0 dx 2x t + 1 Z 4x2 ! Z e2x ! d d (iii) ln t dt (iv) ln t dt dx x dx √ e4 x Z x2 t ! d e (v) √ dt dx x t 2.3. Compute the following integrals. Z π/3 Z 4 2 (i) sec (x)dx (ii) 2x dx π/4 0 2 Z 1 tan(x) Z 3 (iii) (1 − |x| )dx (iv) 2 4 dx −2 −1 1 + x + x 3. You are supposed to know how to use the Substitution Rule to compute the indefinite and/or definite integrals. Example Problems 3.1. Compute the following Z integrals: ln x Z (i) tan x dx (ii) dx x Z Z 0 √ 2 √ (iii) x x + 1 dx (iv) x5 1 + x2 dx Z −1 0 π/4 √ Z π/4 4 (v) sec x tan x dx (vi) 1 + tan x · (sec2 x) dx 0 0 π Z π/2 sin x Z 2 (vii) (1 − sin x) cos x dx (viii) dx π/2√ 0 1 + cos2 x Z ln 3 Z (ln 3−ln 4)/2 ex ex (ix) dx (x) √ dx 0 1 + e2x − ln 2 1 − e2x  5 4. You are supposed to understand that the differential equation dy = ky dx has a solution of the form y = y(0)ekt , and should be able to apply it to analyze the population growth and radioactive decay. In the case of the radioactive decay, you should also understand the formula m(t) = m(0)2−t/h in terms of the half-life h. Example Problems 4.1. A culture of a single cell creature Amoeba is found to triple its population in three weeks. Find its relative growth rate k. The formula for the population P (t) as a function of time t is given as follows using the relative growth rate: P (t) = P (0)ekt. 4.2. The number of bacteria in a cell culture is initially observed to be 50. Three hours later the number is 100. Assuming that the bacteria grow exponentially, how many hours after the initial observation does the number of bacteria become equal to 700 ? 4.3. The half-life of cesium-137 is 30 years. Suppose we have a 120-mg sample at the beginning. How long will it take until the remain of the sample becomes 2-mg ? 4.4. Initially there was 20 gm of a radioactive substance. After 5 hours, only 4 gm remained. What is the half-life of this substance ? 4.5. The population of a town with population of 90, 000 in yera 2016 grows at a rate of 2.4 % per year. After how many years will the population reach 120, 000 ? 6 Materials up to Exam 3 (including Exam 3) Remark: There will be 20 questions regarding the materials up to Exam 3 (including Exam 3) in the Final Exam. Look also at the “NOTE” for the subject matter 14 about computing the total distance. 5. You are supposed to be able to solve the equations involving the trigonometric functions and find solutions on the given interval, using the basic formulas of the trigonometric functions (e.g., double angle formulas for sine and cosine  sin 2x = 2 sin x cos x cos 2x = cos2 x − sin2 x = 1 − 2 sin2 x = 2 cos2 x − 1 and sin2 x + cos2 x = 1, etc.). Example Problems 5.1. How √ many solutions are there on the interval [0, 2π] for the equa- tion 3 sin x = sin(2x) ? 5.2. Find all the values x on the interval [0, 2π] satisfying the equation cos(2x) − sin(2x) = 0. 5.3. Find all the values x on the interval [0, 2π] satisfying the equation cos(2x) − sin x = 0. 5.4. Find all the values x on the interval [0, 2π] satisfying the equation sin2 x + sin x = cos2 x. 6. You are supposed to know how to compute the derivative of a function of the form y = f (x)g(x). Example Problems 6.1. Find the √ derivative of the following functions. (i) y = ( x)tan x (ii) y = xcos x 2 (iii) y = xln(x ) (iv) y = (ln x)sin x 7. TWO “Related Rates” problems will be given in the Final Exam. Of particular importance are: Street light problem Sand pile Problem Ladder problem Water tank problem Area (of a rectangle or a triangle) problem Water drip problem Baseball problem  7 7.1. (Street Light) A person of 6 feet in height is walking towards a streetlight along a straight path at a rate of 3 feet per second. The height of the streetlight is 18 feet above the ground. What is the rate in feet/second at which the person’s shadow is shortening? WARNING: We are asking for the rate of change for the length of his shadow, NOT for the rate of change for the distance from the tip of the shadow to the bottom of the streetlight. 7.2. (Sand Pile) Sand is being dumped from a conveyor belt at the rate of 30 ft3 /min and its coarseness is such that it forms a pile in the shape of a cone whose base diameter and height are always equal. How fast is the height of the pile increasing when the pile is 10 ft high ? 7.3. (Water tank with leak) A leaky water tank has the shape of an inverted cone. The tank is 16 m deep and 4 m in radius at the top. Water pours into the tank at a rate of 10 m3 /min. At the moment when the water in the tank is 12 m deep, the water level is increasing at 31 m/min. Assume that the tank leaks at a constant rate. What is the rate of leakage from the tank ? 7.4. (Ladder: speed) A ladder 10 ft long rests against a vertical wall. If the bottom of the ladder slides away from the wall at a rate of 1 ft/sec, how fast is the top of the ladder sliding down the wall when the bottom of the ladder is 6 ft from the wall ? 7.5. (Ladder: angle) A 15-foot ladder is leaning against a vertical wall and its bottom is being pushed toward the wall at the rate of 2 ft/sec. At what rate is the acute angle θ between the ladder and the ground changing when the acute angle the ladder makes with the ground is π/4 ? 7.6. (Distance) Two people start from the same point at the same time. One walks north at 2 mi/h and the other walks west at 4 mi/h. How fast is the distance between them changing after 30 minutes ? 7.7. (Water drip) Water is falling on the surface, wetting a circular area that is expanding at a rate of 9 mm2 /sec. How fast is the radius of the wet area expanding when the radius is 135 mm ? 7.8. (Baseball) A mini-baseball diamond is a square ABCD with side 9 m. A batter hits the ball at A and runs towards first base B with a speed of 2 m/sec. At what rate is his distance from third base D increasing when he is two-thirds of the way to first base ? 8. You are supposed to understand the idea of the linear approximation of a function f (x) at x = a f (x) ≈ L(x) = f (a) + f 0 (a)(x − a), and apply it to approximate the value of a function at a given point. 8 Example Problems √ 8.1. Find the formula for the linear √ approximation to f (x) = x at a = 9 and use it to approximate 9.3. √ 8.2. Find the formula for the linear √ approximation to f (x) = 3 x at a = 27 and use it to approximate 3 24. √ 8.3. What is the result√of approximating 4 16.32 using the linear ap- proximation of f (x) = 4 16 + x centered at a = 0 ? 8.4. Use the linear approximation of the function f (x) = ln(1 + x) at a = 0 to find the estimate of ln(1.01). 8.5. Suppose y = f (x) is a differentiable function (defined everywhere) that satisfies f (2) = 3 and f 0 (2) = −4. Find the estimate of f (1.97) using the linear approximation of f at a = 2. 9. TWO Optimization Problems will be given in the Final Exam. Of particular importance are: Norman window problem Area (enlosed by the tangent in the 1st quadrant, of a right triangle with ffixed hypotenuse, of an inscribed rectangle) problem Screen problem Hinge problem Box problem Example Problems 9.1. A Norman window is a window in the shape of a rectangle sur- mounted by a semicircle whose diameter equals the base of the rec- tangle. What is the largest possible area of a Norman window with perimeter 8 m ?  9 9.2. What is the smallest possible area of a triangle formed by the x2 y2 coordinate axes and a line tangent to the ellipse 2 + 2 = 1 in the 7 3 first quadrant ? 9.3. What is the largest area of a right triangle whose hypotenuse has lengh 5 ? 9.4. What is the maximum area of a rectangle whose base is on the x-axis, having the remaining two vertices on the graph of y = 9 − x2 , and lying above the x-axis ? 9.5. An auditorium with a flat floor has a large screen on one wall. The lower edge of the screen is 5 feet above eye level and the upper edge of the screen is 20 feet above eye level. How far from the screen should you stand to maximize your viewing angle ? DEF 20 FED 5 q DEF x 9.6. Two wooden bars of equal length AO = BO = 1 ft are connected by a hinge at point O so that one can rotate the bar BO around as shown in the picture below. Find the maximum area of the triangle 4ABC when 0 < θ < π. B θ A O C 9.7. A rectangular box has to be made with the width being twice as long as the length (with a bottom but WITHOUT a top). If the surface area of the box is 400 cm2 , what is the height of the box with the largest volume ? 10 10. You are supposed to understand the notions of continuity and differentiability, and their difference. Example Problems 10.1. Determine whether the followig function is continuous / differen- tiable at the given point. x2 if x < 0 (i) f (x) = at x = 0. x3 if x ≥ 0.  4x if x < 1 (ii) g(x) = at x = 1. x3 + 3 if x ≥ 1 10.2. Find the real numbers a and b so that the following function becomes continuous over (−∞, ∞):  a if x ≤ 1 √  f (x) = 2  12( x + 8 − b) if x > 1. x2 − 1 10.3. Find the real numbers a and b so that the following function becomes continuous over (−∞, ∞):   a if x ≤ 3 f (x) = | − 5x + 5b|  if x > 3. x−3 11. Given a function which is one-to-one, with a given domain and range, you are supposed to be able to find the formula for its inverse function, describing its domain and range. You should also understand the relation between the graph of the original function and the one for its inverse. Example Problems 11.1. Given the following function with the prescribed domain, find the formula, domain and range of its inverse function. 1 − 2x (i) f (x) = defined over (3, ∞) x−3 −3x + 2 (ii) f (x) = defined over (−∞, 1) x−1 6x − 1 (iii) f (x) = defined over (1, ∞) 2x +x 1 e (iii) f (x) = x defined over (−∞, ∞) 2e √ +1 (iv) f (x) = 1 − x + 1 defined over [−1, ∞)  11 12. You are supposed to understand the method of Implicit Differen- tiation to compute the derivative. For example, you should be able to determine the equation of the tangent line to the graph of a function implicitly defined, computing the derivative using the implicit differ- entiation. Example Problems 12.1. Suppose that f is a differentiable function defined on (−∞, ∞) satisfying the equation f (x)+x2 (f (x))3 = 10 and the condition f (1) = 2. Find f 0 (1). 12.2. Find the equation of the tangent line to the curve given by the equation x2 + y 2 + 6x + 3y − 5 = 0 at the point (1, −1). 12.3. Find the slope of the tangent to the curve given by the equation x2 sin(2y) = y 2 cos(2x) at a point (π/4, π/2). 12.4. Find the slope of the tangent to the curve given by the equation 2xy + π sin(xy) = 2π at a point (π/2, 1). dy 12.5. Find given ex/y = 7x − y. dx 13. You are supposed to be able to use the chain rule properly and precisely, even when the function is obtained as the composition of several functions. You are also supposed to know the relation between the derivative of the original function and the derivative of its inverse (when it exists). Example problems 13.1. Compute the derivative of the following function:  9 t−2 (i) y = sin(sin(sin x)) (ii) y = 2t + 1 √ q p (iii) y = x + x + x (iv) y = ln | sec(3θ) + tan(3θ)| 13.2. Suppose that F (x) = f −1 {g(x)}2 and that the functions f
(which is one-to-one, and hence has its inverse) and g satisfy the fol- lowing conditions:   f (2) = 9, f (9) = 5, f 0 (1) = 4, f 0 (2) = 3, f 0 (3) = −2  g(1) = 3, g 0 (1) = 2 Find F 0 (1). 13.3. Suppose that g is a function with g 0 (1/2) = 5. Let f be the function defined by f (x) = g(2 cos2 (x)). Compute f 0 (π/3). 12 14. Given the position function of a particle, you are supposed to be able to compute its velocity, acceleration, understanding its physical meaning. You should be able to determine when a particle is speeding up or down, whether it is accelerating or decelerating. You are also supposed to be able to compute the “total” distance traveled during the given period. NOTE: The last part of computing the total distance requires inte- gration, which was discussed after Exam 3. However, this subject is classified as an item in “Materials up to Exam 3 (including Eam 3)” by convention. Example Problems 14.1. The position of a particle is given by the function s = f (t) = t3 − 6t2 + 9t. Find the total distance traveled during the first 6 seconds. 14.2. The velocity function, which is the 1st derivative of a position function f (t) is given by the following formula v = f 0 (t) = 6(t−1)(t−3). Can you determine the total distance traveled during the first 5 seconds ? 14.3. A rock is thrown upward so that its height (in ft) after t seconds is given by h(t) = 48t − 16t2. What is the velocity of the rock when its height is 32 ft on its way up ? 15. You are supposed to be able to use the 1st Derivative Test, as well as the 2nd Derivative Test, to find the local maximum and local minimum of a function. You are also supposed to find the inflection points by analyzing the behavior of the 2nd derivative. Example Problems 15.1. The first derivative of a function f is given by f 0 (x) = (x + 2)2 (x + 1)(x − 1)3 (x − 3)2 (x − 5). Find the values of x for which the function f takes (a) local maximum, and (b) local minimum. 15.2. Consider the function f (x) = x8 (x − 4)7. (a) Find the critical numbers of the function f. (b) What does the Second Derivative Test tell you about the behavior of f at these critical numbers ? (c) What does the First Derivative Test tell you that the Second Derivative test does not ? 15.3. How many inflection points does the graph of the function y = f (x) = x5 − 5x4 + 25x have ? 15.4. We have a function whose first derivative is given by the formula f 0 (x) = (x − 1)5 (x + 1)5. Find the x-coordinates of the local extrema and the inflection points of the function.  13 16. You are supposed to be able to sketch the graph of a function by computing the 1st derivative (increasing or decreasing) and 2nd derivative (concave up or down). You are also supposed to be able to determine the horizontal/vertical/slant asymptotes of the graph. Example Problems 16.1. Draw the graph of the following function: x3 x2 (a) y = f (x) = 2 (b) y = f (x) = 2 x − 16 x − 16 x 1 (c) y = f (x) = 2 (d) y = f (x) = 2 x − 16 x − 16 x (e) y = f (x) = 2 (f) y = f (x) = ln |x2 − 10x + 24| x + 16 16.2. Find the equations of the horizontal, vertical, and slant asymp- totes of the function 2x3 − 4x2 + 5x − 10 f (x) =. x2 + x − 6 17. You are supposed to know how to find the absolute maximum and absolute minimum of a function f defined on the closed interval [a, b], by comparing the values on the end points f (a), f (b) and the values on the critical value(s) f (c)(0 s). You should know what the definition of a critical value is. Example Problems: 17.1. Find the absolute maximum and absolute minimum values of the function f on the given interval. (a) f (x) = x2/3 on [−1, 8] cos x (b) f (x) = on [−π/2, π/2] 2 − sin x (c) f (x) = 2x3 + 3x2 − 12x on [−1, 2] (d) f (t) = 2 cos t + sin 2t on [0, 2π] (e) f (x) = 4 cos3 (x) + 21 cos2 (x) − 24 cos(x) on [−π/2, π/2]. (f) f (x) = e−x · sin x on [0, 3π/2] 18. You are supposed to know the statement of the Mean Value The- orem as well as its meaning, and also to know under what conditions you can apply the Mean Value Theorem. You are also supposed to be able to know how to apply this corollary of the Mean Value Theorem to compute some value which is seemingly difficult to determine other- wise: If f 0 (x) = 0 for all values of x ∈ (a, b), then a continuous function f on the closed interval [a, b] is actually a constant. 14 Example Problems 18.1. Consider the function f (x) = x4 −2x2 +5x+3 over the interval [−2, 2]. How many values of c ∈ (−2, 2) satisfy the statement of the f (2) − f (−2) Mean Value Theorem: f 0 (c) = ? 2 − (−2) 18.2. Consider the function f (x) = x(x4 −5) over the interval [a, b] = [0, 2]. What value(s) of c ∈ (a, b) satisfies (satisfy) the statement of the f (b) − f (a) Mean Value Theorem: f 0 (c) = ? b−a 18.3. Consider the function y = f (x) = x2/3 over the interval [−1, 1]. Does it satisfy the conditions for the Mean Value Theorem to hold ? f (1) − f (−1) Do we have any value c ∈ (−1, 1) such that f 0 (c) = ?   1 − (−1) 1 18.4. Determine the exact value of tan−1 + tan−1 (7). 7 18.5. Consider the funtion   −1 −1 1 f (x) = tan (x) − tan −. x (i) What can oyu say about f 0 (x) ? (ii) What is the conclusion derived from (i).   −1 −1 1 (iii) Determine the value of f (7) = tan (7) − tan −.  7 1 Determine the value of f (−7) = tan−1 (−7) − tan−1. 7 (iv) Do we have f (7) = f (−7) ? If not, does it contradict (i) and (ii) ? 19. You are supposed to be able to compute the (right/left hand side) limit, understanding its proper meaning, and using the Squeeze Theorem. You are also supposed to be able to determine the exact value of the limit who has an indeterminate form (e.g. 0/0, ±∞/±∞, ∞−∞) using some proper technique.  15 Example Problems 19.1. Compute the following limits: x2 − 9 x2 − 4x − 5 (i) limx→3 2 (ii) limx→5− x − 7x + 12 |x − 5| ! √ 2 5x + 9 − 3 (iii) limx→( π2 )+ etan x (iv) limx→0 x2 √   5 5
(v) limx→0 2 + (vi) limx→∞ x2 − x4 + 16x2 x −x x  3 x−8 (vii) limx→∞ (sin x) · tan (viii) limx→8 √ x 3 x−2 x3 + 3x + 2 x3 + 3x + 2 (ix) limx→∞ √ (ix∗ ) limx→−∞ √ 2x3 + 9x6 + 4x + 5 2x3 + 9x6 + 4x + 5 20. You are supposed to know how to compute the limits using L’Hospital’s 0 ±∞ Rule, under the provision that the limits are formally of the form ,. 0 ±∞ Example Problems 20.1. Compute the following limits: ln(x) 1 + cos x (a) limx→∞ √ (b) limx→π x (x − π)2 2 sin(x ) 7x − 6x (c) limx→0 (d) limx→0 x arctan(cos x − 1) 3 − 2x sin x − x tan x − x (e) limx→0 3 (f) limx→0 x x3 sin x (g) limx→0+ 1 − x2 21. You are suppose to know how to compute the limits of the inde- terminate form ±∞ × 0, ∞ − ∞. Example Problems 21.1. Compute the following limits: 2 tan−1 (x)    (a) limx→0+ sin(x) · ln(2x) (b) limx→∞ x · ln √ π
(c) limx→ π − [(2x − π) · tan x] (d) limx→∞ x2 + 3x + 5 − x ( 2 )  1 1 (e) limx→0+ − x sin x 16 22. You are supposed to be able to compute the limits lim [f (x)]g(x) x→a formally of the form 00 , ∞0 , 1∞. Example Problems 22.1. Compute  the  following limits:  7x 3x+1 3 2x + 1 (a) limx→∞ 1 + (b) limx→∞ x 2x − 1 (c) limx→∞ (2x + e5x )1/x (d) limx→0+ [tan(5x)]sin x 6  x  x−3 7/x (e) limx→0 (1 + 3 sin x) (f) limx→3+ cot x 3 (g) limx→0+ (1 + 3x)