CALCULUS AB WEEKLY REVIEW SEMESTER 2

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Phoebe Hutchinson
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1 CALCULUS AB WEEKLY REVIEW SEMESTER 2

2 This packet will eventually have 12 worksheets. There are currently 5 worksheets in this packet. As the semester progresses, I will add more sheets to this packet. Each worksheet contains 10 quesaons that help keep all topics covered in Calculus AB fresh in students minds. I give my students one of these worksheets on Monday and it is due on Friday. I count each sheet as 10 points toward their homework grade. These are basic quesaons designed to help students remember all of the topics. Topics covered are things such as: finding derivaaves, wriang equaaons of tangent lines, its, opamizaaon problems, related rates problems, curve sketching, finding anaderivaaves, approximaaons using Riemann sums, approximaaons using the trapezoidal rule, solving differenaal equaaons, u- subsatuaon, AND MORE! Graphic on Atle page by: Krista Wallden CreaAve Clips hxp:// Wallden Whimsy Clips hxp://

3 CALCULUS WEEKLY REVIEW #1 1. Find the equaaon of the tangent line to the curve y = 3x 6 at the point where x = 0. 2x Find the derivaave of: y = sin 2 3x 3. What is the it? tan h 0 π 4 + h 1 h 4. Where does the funcaon have a maximum over the interval [0,5]? f(x) = 2x 2 3x Using the same funcaon as in #4, what is the minimum value of the funcaon over all real numbers? 6. Find the derivaave of x 2 3xy + y 2 = Find the it: x 3x 3 + 2x + 5 2x 3 + 4x 3 8. The posiaon of a paracle is given by the equaaon s(t) = 2t 3 24t t 7. Find the equaaon for the velocity and the acceleraaon of the paracle. 9. Using the same situaaon as in #8, when does the paracle change direcaon? 10. Using the same situaaon as in #8, how far did the paracle travel between 0 and 3 seconds?

4 CALCULUS WEEKLY REVIEW #2 1. If f (x) = 3x 2 + 6x + 5 and f(2) = 5, find f(x). 2. What value of c is guaranteed by the Mean Value Theorem over the interval [0, 3] for the funcaon f(x) = x Write the equaaon of the tangent line to the funcaon y = sin x at the point where x = π. Write your answer in slope- intercept form. 4. A paracular cylinder has a radius that is increasing at a rate of 5 cm/s and height that is decreasing at a rate of 8 cm/s. How fast is the volume of the cylinder changing when the radius is 10 cm and the height is 12 cm? 5. If f(x) = 3 + x + 2, then what is f (2)? 6. At what value(s) of x does f(x) = x 4 8x 2 have a relaave minimum? 7. If f(x) = x 4 + bx 2 + 8x + 1 has a horizontal tangent and a point of inflecaon for the same value of x, what does b have to be? 8. Suppose that f (x) = x 3 x 2. Over what interval is the graph of f(x) both increasing and concave up? 9. Find the derivaave of y = arctan(2x) 10. Find the equaaon of the horizontal asymptote(s) to the curve: y = 2x2 + 5x 3 3x 2 4x + 2

5 CALCULUS WEEKLY REVIEW #3 1. For what value of k will y = x + k have a relaave minimum at x = 3? x 2. Find the slope of the normal line to the curve y = (2x- 3) 2 at the point where x = What is the it? sin h 0 3π 2 + h +1 h 4. Using 3 right hand rectangles, approximate the area between the curve y=x 2 and the x- axis over the interval [0, 3]. 5. Using the same situaaon in #4, use 3 leo hand rectangles to approximate the area. 6. Find the derivaave of x 2 + xy + y 2 = Using the same equaaon as in #6, find the coordinates where the curve has a veracal tangent line. 8. A conical tank 20 feet in diameter and 30 feet tall is leaking water at a rate of 5π cubic feet per hour. At what rate is the water level dropping when the water is 15 feet deep? (Show a complete soluaon on the back.) 9. What is the minimum value of the funcaon y = 3x 2-4x - 5 over the interval [- 1 5] 10. Find the area between the curve y = x 3-4x 2-7x+10 and the x- axis over the interval from [0, 1].

6 CALCULUS WEEKLY REVIEW #4 1. Find the equaaon of the tangent line to y = (3x +4) 3 at the point where x = Two cars start at the same place and at the same Ame. One car travels west at a constant velocity of 80 miles per hour and a second car travels south at a constant velocity of 60 miles per hour. Approximately how fast is the distance between them changing one- half hour later? (Show work on the back) 3. What is the it? x 3 27 x 3 x 3 4. Find the area between the curves f(x) = x 2 +2x+1 and g(x) = x Find the equaaon of the tangent line to the graph of y = x 3 + 3x at its point of inflecaon. 6. Find the approximaaon of the area under the curve using 4 trapezoids. Use the funcaon y = x 2-4x+4 over the interval [0, 4]. 7. Find the value of k if 8. Find the derivaave of: 3 3x 2 + k dx = 33 y = arcsin x A rectangular box with a square base and no top has a volume of 500 cubic inches. Find the dimensions of the box that require the least amount of material. (Show work on the back.) 10. Find the derivaave of: y = sin 3 x

7 CALCULUS WEEKLY REVIEW #5 1. Find the area in the first quadrant that is enclosed between the graphs of f(x) = x and g(x) = x Find the derivaave of: y = sec 3 2x 3. What is the it? sin h 0 π 3 + h 3 2 h 4. A funcaon is quadraac and passes through the point (2, 9) and (0, 1). The slope of the graph is 6 at x = 2. Find the funcaon. 5. The funcaon f(x) is conanuous for all real numbers. f (x) = x2 7x +12 x 3 What is f(3)?, x If 3 f (x) = 3x, what is f (8)? 7. The length of a rectangle increases by 3 feet per minute while the width decreases by 2 feet per minute. When the length is 15 feet and the width is 40 feet, what is the rate at which the area is changing? 8. If x 2-2xy+3y 2 = 8, find the derivaave. 9. Find the integral use u- subsatuaon. x 3x 2 8 dx 10. The acceleraaon of a paracle moving along the x- axis at Ame t is a(t) = 4t 6. The velocity of the paracle at t = 1 is 5 and the posiaon of the paracle at t = 0 is 4, find the posiaon funcaon.

8 CALCULUS WEEKLY REVIEW #6 1. Evaluate at x = π. sec 3(x + h) sec(3x) h 0 h 2. What is the absolute maximum value of y=x 3 3x over [- 2, 4]. 3. Solve the differenaal equaaon. f (x) = 3x 2 + 2x + 4, f (2) = - 1, f(0) = 3 4. What is the maximum acceleraaon of a paracle over [0, 3] if the velocity of the paracle is v(t)=t 3 3t t + 4? 5. Find the second derivaave when x = 3. x 2 + y 2 = If a snowball decreases in volume at the rate of 4π cm 3 /min, how fast is the radius changing when the radius equals 2? 7. Find the derivaave of y = 7 x. 8. Evaluate: a) ln 1 = b) e 0 = c) ln e = 9. Find the equaaon of the tangent line the curve at the point where f (x) = 2. f (x) = (2x + 3) Find the area between the curve y = 2x 3 5x 2 x + 6 and the x- axis over [- 1, 2].

9 CALCULUS WEEKLY REVIEW #7 1. Find the it. 2. For what value of a is the funcaon conanuous? 3x 4 + 2x ax 2 1, x < 2 x (x 2 + 2)(5x 2 +1) f (x) = x 3 +1, x 2 3. If f(x) = (x 2 )(sin x), what is the value of f (π)? 4. What is the x- coordinate of the point of inflecaon for f(x) = 4x 3 + 3x 2 6x? 5. Find the equaaon of the tangent line to 6. The posiaon of a paracle is given by: at the point where x = 1. x y = Write your answer in s(t) = 1 1+ x 2 standard form. 3 t t 2 2t +1 How far has the paracle traveled in the first 4 seconds? 7. ConAnuing with #6 what is the maximum velocity of the paracle over the interval [0, 4]? 8. The volume of a cube is shrinking at a rate of 25 cm 3 /min. How fast is the surface area changing when the edge of the cube is 20 cm? 9. Evaluate the integral. Show your work. 2 0 x 2 (2 + x 3 )dx 10. Find the area between the x- axis and the curve f(x) = x from [0, 3].

10 CALCULUS WEEKLY REVIEW #8 1. Find the equaaon of the tangent line to the curve y = (ln x) 2 at x = Find the area between the curves y = x 2-4 and y = x What is the it? x 2 2x x + 2 g(x) = 4. Find the values of a and b so the funcaon is always conanuous. 3, x 2 ax + b,2 < x < 5 2, x 5 5. The area of an equilateral triangle is increasing at 10 cm 2 /min. Find the rate at which the height is changing when the area is Consider the integral: Use the subsatuaon u = x 2 Rewrite the integral completely in terms of u. (Don t forget to switch the its.) x 2 x 2 dx 7. What is the average value of y=x 2 over [0, 3]? 8. Evaluate. 2 d t dx dt x 2 9. Draw the slope field for dy dx = x y 10. A paracle moves along the x- axis according to x(t)=1 sin(2t). Find where the velocity of the paracle equals zero over the interval from [0, π].

11 CALCULUS WEEKLY REVIEW #9 1. Find the its. A) f (x) x 1 2. What is the slope of the curve when y = - 4? x 2 xy = 5 B) f (x) x 0 C) f (x) x Find the derivaave. Simplify so your answer is wrixen as one fracaon. e x y = ln e x Use a leo hand Riemann sum with 5 rectangles to esamate the area between the x- axis and the curve y = x 2 over the interval from [- 3, 2]. What is the difference between the actual area and the area you found by using the leo hand Riemann sum? (Show work on the back) 5. Find the equaaon of the normal line to the curve y = (x+2) 3 when x = A paracle is moving along a curve according to the equaaon s(t) = 3t(t- 2) 3. Consider the moaon of the paracle over the interval [1, 4]. At what Ame is the average velocity of the paracle equal to the instantaneous velocity of the paracle? (Use your calculator!) 7. sinθ cos 2 θ dθ 8. Where is the funcaon decreasing? y = x 4-4x 3 +4x Find the volume of the solid that would be formed if you rotated the area in the first quadrant between the curve y = - x+1 and the x- axis around the x- axis. 10. Find the derivaave of: y = arctan x 3

Set up the it definiaon for finding the derivaave of y = sin x. 4. Find the values of a and b so the funcaon will have a relaave extreme at (3, 1). y = 1 3 x3 + ax 2 + b 5.

12 CALCULUS WEEKLY REVIEW #10 1. For what value of k does the it exist? x 2 + 3x + k x 2 x 2 2. A streetlight is 15 feet tall. A boy who is 6 feet tall is walking away from the light at a rate of 5 feet/s. Determine the rate at which his shadow is lengthening at the moment he is 20 feet from the light. SHADOW 3. Set up the it definiaon for finding the derivaave of y = sin x. 4. Find the values of a and b so the funcaon will have a relaave extreme at (3, 1). y = 1 3 x3 + ax 2 + b 5. Find the equaaon of the tangent line to the curve y = x 2 3x at the place where the parabola crosses the y- axis. 6. If g(x) = x 3 + x 2 + 1, find the value of g (2). x 7. If h(x) = 2x 3, what is the value of the derivaave at x = - 2? 8. What is the value of c guaranteed to exist by the Mean Value Theorem if m(x) = x 4 3x over the interval [1, 3]. 9. This is the graph of the velocity of a given paracle moving along the x- axis in meters/ second. At what Ame(s) does the paracle reverse direcaon? 10. ConAnuing #9, how far did the paracle move during the 10 seconds shown on the graph?

13 CALCULUS WEEKLY REVIEW #11 1. Rotate the space bounded by y = x 2 and y = x 3 in the first quadrant around the y- axis. What is the volume of the solid that is formed? 2. Find the derivaave of: y = (x)(sin x 2 ) 3. What is the slope of the curve y 3 y = 3x + 3 at the point where x = 1? 4. If you have a 10 foot ladder that slides down a wall at 2 feet per minute, how fast is the boxom of the ladder sliding when the top is 8 feet above the ground? 5. On what interval(s) is f(x) = x 3 + 6x 2 15x + 2 increasing? x 6. If g(x) = t 4 dt, what is g (3)? 0 7. Find the volume of the solid whose base is bounded by y = x 2, the x- axis, and the line x = 2. The cross secaons of the solid are squares that are perpendicular to the x- axis. 8. Use your calculator to find the slope of the line tangent to y = e - x at x = A paracle moves along a line according to the equaaon s(t) = 4 cost t What is the velocity of the paracle when the acceleraaon is equal to zero? Only consider the interval [0, π]. (Use your calculator.) 10. Find the area between the curves y = 5x x 2 and y = 2x in the first quadrant.

AP Calculus BC Chapter 4 AP Exam Problems A) 4 B) 2 C) 1 D) 0 E) 2 A) 9 B) 12 C) 14 D) 21 E) 40

AP Calculus BC Chapter 4 AP Exam Problems A) 4 B) 2 C) 1 D) 0 E) 2 A) 9 B) 12 C) 14 D) 21 E) 40 Extreme Values in an Interval AP Calculus BC 1. The absolute maximum value of x = f ( x) x x 1 on the closed interval, 4 occurs at A) 4 B) C) 1 D) 0 E). The maximum acceleration attained on the interval