Transcription

1 Resources:

2 PCTI Mathematics Department Sal F. Gambino, Supervisor Summer Packet Grading On the first day of school, the teacher will check for completion/effort of the packet. This will be weighted at 50%. Teacher will then review the packet with the students. Upon completion of the review, the students will be given an assessment based on the summer packet. The assessment will be weighted at 50%. The two weighted scores combined will count as one project grade. Therefore, the grade for the summer packet will be placed under the project category.

3 . Use the definition of a derivative to find y of y = 2 t Related Rates # 2 #3 2. A swimming pool is meters long, 6 meters wide, meter deep at the shallow end, and 3 meters deep at the deep end (see the figure above). Water is being pumped into the pool at cubic meter per minute, and there is meter of water in the deep end. a.) What percent of the pool is filled? b.) At what rate is the water level rising? 3. A trough is 2 feet long and 3 feet across the top (see figure). Its ends are isosceles triangles with altitude of 3 feet. a.) If water is being pumped into the trough at 2 cubic feet per minute, how fast is the water level rising when the depth h is foot? b.) If the water is rising at a rate of 3 inch per minute when h = 2, determine the rate at which water is being pumped into the trough.

4 Solve. (5x 3 7x)dx 5. (sec(2x) tan (2x))dx 2 6. (x 3) /2 dx (x ) 2 xdx. ( 2x 9 sinx)dx 9. xsin3x 2 dx Sketch region bounded by graph and find area of regions. 0. y = x 2, y = x + 2. f(y) = y 2, g(y) = y g(x) =, y =, x = 0 2 x 3. Definef(x) = x x (a) Show that f(x) is continuous at x = 2. (b) Where on the interval [-2,2] is f discontinuous? Show the work that leads to your conclusion. (c) Classify the discontinuities in part (b) as removable or nonremovable. Find the limit (if it exists). If the limit does not exist, explain why. (x + y) 3 x 3. lim y 0 y 5. lim(x 2)(x 2)2 x 6

5 cosx 6. lim x 0 sinx 2x 2 7. lim x 3x lim x ( + x ) Free-Falling Object, for 9 and 20 use the position function s(t) =.9t which gives the height (in meters) of an object that has fallen from a height of 250 meters. 9. Find the velocity of the object when t =. 20. At what velocity will the object impact the ground? Find the one sided limit (if it exists) sinx 2. lim x 0 5x cos 2 x 22. lim x 0 x x lim x 3 x 3 2. Find the equation of the tangent to the graph of f(x) = 3x 5cos2x at x = Suppose g(0) =, g (0) =, and g (0) = 2. If h(x) = g(x). What is h (0)? Determine relative max or min, and state on which intervals function is increasing or decreasing 26.f(x) = x + x 2

6 27. f(x) = sin(x) +, 0 < x < 2π 2. The concentration C of a chemical in the bloodstream t hours after injection into muscle tissue is C(t) = 3t 27 t3, t 0. Determine the time when the concentration is greatest. 29. R is a region bounded by f(y) = y 2 3 and g(y) = 3y +. Write the equation that gives the area of R. 30. The functions f(x) = 2 x 2 and g(x) = x are given. a. Find the area of the region bounded by the graphs of f(x) and g(x). b. Find the area of the region bounded by the graph of f(x) and the x-axis. c. Find the area of the region bounded by the graph of g(x), the x-axis, and the line x=2. Determine whether the Mean Value Theorem can be applied. If so, find all values of c on the open interval (a,b) such that f (c) = f(b) f(a). If not, explain why not. b a 3. f(x) = x 2 3, [, ] 32. f(x) = x cos x, [ π 2, π 2 ] Use the First Derivative Test to find any relative extrema of the function. 33. f(x) = x 3 5x 3. g(x) = 3 sin (πx ), [0, ] 2 2

7 35. Determine the points of inflection and discuss the concavity of the graph of f(x) = (x + 2) 2 (x ) 36. Find the particular solution of the differential equation f (x) = 6(x ) whose graph passes through the point (2, ) and is tangent to the line 3x y 5 = 0. Use properties of summation and evaluate the sum (i + ) i= (2i i 2 ) i= 39. Use left and right-hand sums to approximate the areas of the region using the indicated number of subintervals of equal width y = 0 x 2 +

8 0. Use the limit process to find the area of the region between the graph of the function and the x-axis over the given interval. y = 5 x 2 [-2,]. f(x) = 2x Set up a definite integral that yields area of the region. 2. Properties of definite integrals: Given f(x)dx = 2 Evaluate (a) [f(x) + g(x)]dx (b) [f(x) g(x)]dx (c) [2f(x) 3g(x)]dx (d) 7f(x)dx and g(x)dx = Use the Fundamental Theorem of Calculus to evaluate the definite integral (x x) dx. Sketch the region bounded by the graphs of the functions. Then find its area. y =, y = 0, x =, x = 9 x 5. Use the Second Fundamental Theorem of Calculus to find F (x) F(x) = (t 2 + 3t + 2)dt x 3

9 6. Find the average value of the function over the given interval. Find the values of x at which the function assumes its average value. F(x) = x 3, [0,2] 7. Define f(x) = x, 2 a.) Use the Trapezoidal Rule with n = 5 to estimate f(x)dx. b.) f(x) = ln x + C. Use the Fundamental Theorem of Calculus to calculate the exact value 2 of f(x)dx. c.) Explain why the Trapezoidal Rule cannot be used to estimate f(x)dx. Multiple-choice questions. Let f(x) = (3 + 2x x 2 ) 3 be defined for the closed interval 2 x 3. If M is the y- coordinate of the absolute maximum and m is the y-coordinate of the absolute minimum, what is the absolute value of M+m? a) 9 b) 25 c) 6 d) 6 e) none of these 9. A square is inscribed in a circle. How fast is the area of the square changing when the area of the circle is increasing at the rate of in 2 /min? a) in2 min b) in 2 2 min c) 2 in 2 π min d) in2 min e) π in 2 2 min

10 50. What is the slope of the line tangent to the curve y 3 + x 2 y 2 3x 3 = 9 at the point (,2)? a) 6 b) c) d) e) Find 3x 2 f(x 3 )dx if f(t)dt = k a) k 3 b) 9k c) 3k d) k e) k 3 x If F(x) = t + 3dt, what is F (x) a) x b) 2 X 2 +3 c) 2x( x 2 + 3) d) 2(X2 +3) e) None of these

11 53. Let R be the region between the function f(x) = x 3 + 6x 2 + 0x+, the x-axis, and the lines x = 0 and x =. Using Trapezoidal Rule, compute the area when there are equal subdivisions. a) 96 b) 2 c) 296 d) 396 e) None of these 5. Which of the following statements about the function given by f(x) = x 2x 3 is true? a) The function has no relative extremum. b) The graph of the function has one point of inflection and the function has two relative extrema. c) The graph of the function has two points of inflection and the function has one relative extremum. d) The graph of the function has two points of inflection and the function has two relative extrema. e) The graph of the function has two points of inflection and the function has three relative extrema. 55. The area of the region in the first quadrant between the graph of y = x x 2 and the x- axis is: a) b) 3 c) 2 2 d) 2 3 e) 6 3