MTH Calculus with Analytic Geom I TEST 1

Transcription

1 MTH Calculus with Analytic Geom I TEST 1 Name Please write your solutions in a clear and precise manner. SHOW your work entirely. (1) Find the equation of a straight line perpendicular to the line 4x 7y = 3 and passing through the point (3, 5). (2) Complete the square and find the minimum or maximum value of the quadratic function y = 4x 2 12x + 7 (3) Given the functions f(x) = 3 x and g(x) = 1 x 2, find the composite functions a). (fog)(x) and its domain 1

2 b). (gof)(x) and its domain (4) Find cos θ, tan θ and sec θ if sin θ = 3 5 and π 2 θ < π. (5) What is the domain of the function f(x) = x 1 x 4? (6) Given that f(x) = 4 2x x+1, find (a). the domain of f(x) 2

3 (b). the range of f(x). (7) Without using calculator, evaluate tan ( cos ) (8) Solve for x if a). 9 2x = ( 1 3 ) 6 x b). 5 x2 3x = 1 25 (9) Simplify a). log

4 b). log log 5 5 c). log d). True or false: log 10 ( 5) = log 10 5? Why? (10) Estimate the limits a). (3x 3 + 2x 2 ) lim x 1 2 b). lim x 4 x 2 5x+4 x 2 x 12 Extra Credit: Find x such that the point (x, 4) lies on the line of slope m = 1 3 through (3, 6). 4

5 MTH Calculus with Analytic Geom I TEST 2 Name Answer 10 questions ONLY. SHOW your work entirely. (1) Evaluate a). lim t 9 t 3 t 9 b). x lim 2 +3x+2 x 2 x+2 (2) Evaluate sin 7θ a). lim θ 0 sin 3θ 1

6 sin 9h b). lim h 0 h (3) Evaluate the limit 3x a). lim 2 +20x x 6x x 2 b). lim 36x 4 +7 x 4x

7 (4) a). Given that f(x) = x 2 + 4, compute f (3) using the limit definition of derivative. b). Given that f(3+h) f(3) h = 5h + 30, find f (3). Hint: use limit definition of derivative. (5) Calculate the derivative of the following using any method. a). f(x) = 6x 5/3 3x x 3 3

8 b). f(x) = 3e x x 6 c). f(x) = 4 x + 1 x (6) Use the graph of f(x) in Figure 1 to answer the following question. Figure 1: Graph of f(x) (a). lim f(x) (b). lim f(x) x 1 x 1 + (c). lim x 1 f(x) 4

9 (d). lim f(x) (e). lim f(x) x 3 x 3 + (f). lim x 3 f(x) (g). lim f(x) (h). lim f(x) x 5 x 5 + (i). lim x 5 f(x) (7) Find the derivative of the following functions. a). f(x) = x2 +1 x 2 1. b). f(x) = 3x 2 e x. 5

10 (8) a). Find the point on the graph of f(x) = (2x + 1)(5x 4) where the tangent line is horizontal. b). Given that the function f(x) satisfies f(2) = 3, f (2) = 1, g(2) = 2, g (2) = 1, calculate (f g) (2) (9) a). Calculate f (0) given that f(x) = x 2 e x 6

11 b). Calculate the first four derivatives of f(x) = x. (10) What is the equation of the tangent line to the graph of f(x) = 1 x at x = 4? (11) Compute the derivative of the following functions: a). f(x) = cos x using the limit definition of derivatives. b). f(x) = x sin x+2 using any method. 7

12 (12) Find the derivative of the following functions: a). f(x) = ( 4 2x 3x 2) 5 b) f(x) = 3 x 2 1 8

13 Extra Credit: a) (5 pts). Find the general formula for f (n) (x) given that f(x) = xe x. sin 5x sin 2x b) (5 pts). Evaluate lim x 0 sin 3x sin 4x 9

14 MTH Calculus with Analytic Geometry I - Test 3 Fall 2014 Name Please write your solutions in a clear and precise manner. Answer only 10 questions. (1) (10 pts) Find dy/dx given that y = xy 2 + 2x 2. b). Show that (0, 0) is a point on the curve where the tangent line to the curve is horizontal. 1

15 (2) (10 pts) a). Compute the derivative of the following: a). f(x) = (12x 3 5x 2 + 3x) 10 log 4 (x 2 5x + 7) b). f(x) = (4x 2 + 1) sin x 2

16 (3) (10 pts) Find the derivative of y = [ tan 1 (1 x 2 ) ] 3 + sin 1 (2x) + ( ) 1 x 1 3 3

17 (4) (10 pts) a). Calculate g (b), where g is the inverse of the function f(x) = 3 x , b = f(2). b). Find dy/dx given that y = tan 1 ( x 3 ). 4

18 (5) (10 pts) Given that f(x) = 1 x, without using calculator, find 2 9 a). f, where a = 5, x = b). f(5.01) c). the linearization of the function at x = 5 5

19 (6) (10 pts) Find the equation of the tangent line to the curve xy + x 2 y 2 = 6 at the point (2, 1). 6

20 (7) (10 pts) Without using calculator, a). estimate the approximation b). estimate the approximation cos 1 (0.55) π/3. 7

21 (8) (10 pts) Find the derivative of the following: a). f(x) = x2 (3x 3 +1) (2x+1)(4 5x 2 ) b). f(x) = (x 3 + 1)(x 4 + 2x) 6 (x 5 + 3) 4 8

22 (9) (10 pts) Consider a rectangular tank whose base is 16ft 2. a). How fast is the water level rising if water is filling the tank at a rate of 0.6ft 3 /min? b). At what rate is water pouring into the tank if the water level rises at a rate of 0.8ft/min. 9

23 (10) (10 pts) Find the derivative of the following: a). f(x) = (2x + 5) 12 8x b). g(x) = x x x 4 10

24 (11) (10 pts) Find the point(s) on the graph of x 2 + x y2 = 6 + y where the tangent line is vertical. 11

25 (12) (10 pts) Find the derivative of the following: a). y = sin 1 ( 7x 2 + 2) b). y = cos 1 (1/t) sin 1 ( t). 12

26 Extra Credit (10 pts) Given the graph x n + y n = nx ny + 2, (n 1). a). Show that the graph passes through (1, 1). b). If n = 1, show that the graph is y = 1. c). Show that the tangent line to the graph at the point (1, 1) is horizontal. 13

27 MTH Calculus with Analytic Geometry I - Test 4 Fall 2014 Answer all questions. Write your solutions in a clear and precise manner. Name 1

28 (1) (10 pts) a). Find the minimum and maximum of the function f(x) = x 3 12x x on the interval [0, 2]. b). Verify Rolle s Theorem for the given function and interval: f(x) =, [3, 5]. x2 8x 15 2

29 (2) (10 pts) Given the graph of the polynomial function f(x) Figure 1: Compute the following: a). critical points of f(x) b). local maximum and local minimum value of f(x) c). Absolute Maximum and Minimum value of f(x) on the interval [ 2, 2] d). Intervals where f(x) is increasing and decreasing 3

30 (3) (10 pts) a). Find a point c satisfying the conclusion of the MVT for the function f(x) = x for the given interval [9, 25]. b). Find the value of a and b such that the function f(x) = x 5 bx 2 + ax has inflection point at x = 1 and critical point at x = 2. 4

31 (4) (10 pts) a). Find the critical points of the function f(x) = x 2 + (10 x) 2. b). Find the intervals where the function is increasing or decreasing c). Find the local maximum and/or local minimum 5

32 (5) (10 pts) Let f(x) = x 2 (x 2 4). a). Use Second Derivative Test to find the minimum and maximum value of f b). Find the point of inflection of f(x) c). Find the intervals on which f is concave up or down 6

33 (6) (10 pts) Decide whether or not you can use L Hôpital s Rule to evaluate the following, and then evaluate: sin 4x a). lim x 0 x 2 +3x+1 sin x x cos x b). lim x 0 x sin x c). lim x x+ln x 2x 2 ln x 7

34 (7) Sketch the graph of the function f(x) = x 6 2x 3 using the following steps: a)(1 pt) find the end behavior of f b)(1 pt) find the x and y intercepts of f c)(1 pt) find the maximum and minimum value of f(x) using the second derivative test d)(1 pt) find the interval where f is increasing and decreasing e)(1 pt) find the interval where f is concave down and concave up 8

35 f)(5 pts) use the information from a to e to sketch f(x) 9

36 (8) (10 pts) a). Show (don t find, just show) that the function f(x) = e x 5x has zeros in the intervals [0, 1] and [2, 3] b). With the help of your calculation, find the two exact zeros in the given interval using Newton s Method. 10

37 (9) (10 pts) A cylinder of radius r and height h has surface area 2πr 2 + 2πrh and volume πr 2 h. Find the dimension(s) of a cylinder with surface area 10 and maxima volume. 11

38 (10) (10 pts)show that the equation 1 + x = x 7 has a solution in the interval [1, 2]. With the help of your calculation, use Newton s method to find the exact solution. 12

39 Extra Credit (10 pts) a). Find the derivative of the function f(x) = x sin x. b). Find the derivative of f(x) = (2x + 5) 15 x

40 MTH Calculus with Analytic Geometry I Final Test Fall 2014 Answer any 10 questions. Questions 4 and 10 are compulsory Write your solutions in a clear and precise manner. Name 1

41 (1) (10 pts) If cos θ = 1 3 where 3π/2 θ < 2π a). Without using calculator, find i). sin θ, ii).tan θ b). Given the functions f(x) = 1 x, g(x) = 1 x 2, find i). (fog)(x) ii) (gof)(x) iii). Domain of (f og)(x) 2

42 (2) (10 pts) a). Without using calculator, evaluate sin ( cos 1 2 ) 5 b). Simplify i). 7 log 7 (26) ii). log log 3 3 c). Solve for x given that i). 2 x2 2x = 8 ii). 9 x/2 = ( ) 1 x 1 3 3

43 (3) (10 pts) a). Find the equation of a straight line perpendicular to the line 2x 5y + 3 = 0 and passing through the point (3, 5). b). Complete the square and find the minimum or maximum value of the quadratic function y = 2x 2 + 3x 1. 4

44 (4) (10 pts) Find the derivative of the following functions i). f(x) = 6 3 x 1 2 x ii). f(x) = (2 + sin x)e 4x (iii). f(x) = 7 2x 4 + 5x 2 4 5

45 ( ) 5/3 iv). f(x) = 2x 5 9x 2 +2x v). f(x) = (x 3 1)(x 4 + 2x) 6 (x 5 + 3) 4 6

46 (5) (10 pts) Given the graph of f(x) Figure 1: Compute the following: a). critical points of f(x) b). local maximum and local minimum value of f(x) c). Absolute Maximum and Minimum value of f(x) on the interval [ 2, 3] d). Intervals where f(x) is increasing and decreasing e). Zeros of f(x) 7

47 (6) (10 pts) Let f(x) = x 3 4x 2 + 4x. Find i). the critical point of f(x). ii). the region where f(x) is increasing and decreasing iii). using second derivative test, find the minimum and maximum value of f(x) iv). find the point of inflection of f(x) v). find the region where f(x) is increasing and decreasing. 8

48 (7) (10 pts) Compute the following given the graph of the function f(x) (a). lim f(x) (b). lim f(x) x 1 x 1 + (c). lim f(x) x 1 (d). lim x 2 f(x) (e). lim f(x) (f). lim f(x) x 2 + x 2 + (g). lim x 3 f(x) (h). lim f(x) (i). lim f(x) x 3 + x 3 + (j). lim x 4 f(x) (k). lim f(x) (l). lim f(x) x 4 + x 4 + 9

49 (8) (10 pts) A cylinder of radius r and height h has surface area 2πr 2 + 2πrh and volume πr 2 h. Find the dimension(s) of a cylinder with surface area 15 and maxima volume by answering the following questions i). What is the constraint equation relating r and h? ii). Find a formula for the volume of the cylinder in terms of the radius r iii). Solve the optimization problem. 10

50 (9) (10 pts) a). Use the given graph of y = x 4 + 2x 3 1 and the Newton s Method to find the two exact zero of the function. b). Find the point on the graph of x 2 + x y2 = 2 + y where the tangent line is horizontal 11

51 (10) (10 pts) Evaluate the following i) t5 t 2/3 + 5e 2t 3 dt ii). 2 1 x 3 +3x 4dx x 2 iii). sin(3x + 4) 7 x dx 12

52 b). Use substitution method to evaluate the following i). x 2 x 3 + 4dx ii) x 2 dx 13

53 (11) (10 pts) Evaluate the following limits 3x i). lim 2 4x 4 ( show your work clearly) x 2 2x 2 8 ii). lim 36x 2 +7 x 9x+4. ( show your work clearly) sin 5x (iii). lim x 0 sin 7x ( show your work clearly) 14

54 (12) (10 pts) a). Find dy/dx given that xy 2 + x 2 y 5 x 3 = 1 b). Calculate the second derivative of y = x 2 e 3x 15