MATH 2053 Calculus I Review for the Final Exam

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1 MATH 05 Calculus I Review for the Final Exam (x+ x) 9 x 9 1. Find the limit: lim x 0. x. Find the limit: lim x + x x (x ).. Find lim x (x 5) = L, find such that f(x) L < 0.01 whenever 0 < x <. sin x(1 cos x). Find lim x 0 5. Let f(x) = { c x c sin x x. (x+ x) 6. Find lim (x+ x)+1 (x x+1) x 0. x x π. Find all values for the constant c so that f(x) is continuous for all x. x > π x+5 5. Find lim x 0. x x 8. Find lim +15 x 5. x+5 9. Find lim x 5 csc ( πx ). 10. Find lim x (x + ) then use the definition of a limit to prove your claim. 11. Discuss the continuity of f(x) = 9 x on the closed interval [, ]. 1. Use at least four secant lines to estimate the slope of the line tangent to the function f(x) = x + x at the point (, 5). 1. Find an equation of the tangent line to the graph of f(x) = 1 at the point (, 1). x 1 1. Find the equation of the tangent line to the graph of f(x) = x 1 when x = Find an equation of the tangent line to the graph of f(x) = cos x (cos x + sin x) when x = 0. x x < Let f(x) = { x = 1. Answer the following questions. x > 1 a. Find lim x 1 f(x). b. Find lim x 1 + f(x). c. Evaluate lim x 1 f(x). d. Discuss the continuity of f(x). e. Draw the graph of f(x) to confirm your answers to parts (a) (c). x 1. Let f(x) = 1. Find the following limits without graphing. x 5 a. lim x 5 f(x) b. lim x 5 + f(x) c. Discuss the continuity of f(x). List any intervals over which f(x) is continuous. 18. Use composite functions to evaluate lim x 9 x x +. (Hint: List appropriate choices for f(x) and g(x) such that (f g)(x) = x x +. Then use the theorem for the limit of a composite function.) 19. Use the Intermediate Value Theorem to show that f(x) = x has a zero in the interval [1, ]. 0. Find all points on the graph of f(x) = x x + x at which the slope of the tangent line is. 1. Find dy/ if y = 9 tan x 8 cos x + x.. Find dy/ for y = (x + 1) (x 5x).. Find f (x) if f(x) = x x. x. Differentiate y = 5x 1 x.

2 5. Find f (x) if f(x) = x x. x 6. Find the derivative of y = cot (sin x).. Find the derivative of y = tan (x ). 8. Calculate the derivative of f(x) = x + using the limit definition. 9. Suppose the position equation for a moving object is given by s(t) = t t + 6 where s is measured in meters and t is measured in seconds. Find the velocity of the object when t = The position function for a particular object is s = 1t + 51t + 8. Which statement is true? a. The velocity at time t = 1 is. b. The initial position is 51. c. The velocity is a constant. d. The initial velocity is. e. None of these. 1. Suppose the position equation for a moving object is given by s(t) = 6t + t + 8 where s is measured in meters and t is measured in seconds. a. When does the object hit the ground? b. What is the object s velocity upon impact?. Find the derivative of s(t) = sec t.. Find dy/ for 5x xy + y = 0.. Find dy/ using implicit differentiation for x + x y yx = y. 5. Find dy/ using implicit differentiation for x + y + xy = and then find an equation of the tangent line to the curve at the point (1, 1). 6. Find the second derivative of y = x x+.. A machine is rolling a metal cylinder under pressure. The radius r of the cylinder is decreasing at a constant rate of 0.05 inches per second, and the volume V is 18 cubic inches. At what rate is the length h changing when the radius is.5 inches? [Hint: V = r h.] 8. A metal cube contracts when it is cooled. If the edge of the cube is decreasing at a rate of 0. centimeters per hour, how fast is the volume changing when the edge is 60 centimeters? 9. Determine whether the Mean Value Theorem applies to f(x) = 1 on the interval [, 1 ]. If the x Mean Value Theorem applies, find all values c in the interval such that f (c) = f( 1 ) f( ). If the Mean Value Theorem does not apply, state why. 0. Decide whether the Mean Value Theorem can be applied to f(x) = 6 x on the interval [1, 6]. If the Mean Value Theorem can be applied, find the value(s), c, in the interval such that f (c) = f(b) f(a). If the Mean Value Theorem cannot be applied, state why. 1. Determine from the graph to the right whether the function f possesses extrema on the interval [a, b]. Label any relative extrema on the graph.. Given that f(x) = x + 1x has a relative maximum at x = 6, choose the correct statement: a. f is positive on the interval (6, ). b. f is positive on the interval (, ). b a 1 ( ) c. f is negative on the interval (6, ). d. f is negative on the interval (, 6).. Find all extrema of f(x) = x x on the interval [, ]. Give exact ordered pairs.. Use the First Derivative to list the critical numbers and determine the increasing/decreasing intervals for the function f(x) = x sin x on the interval [0, ]. 5. Give the sign of the second derivative of f at the indicated point for the graph to the right.

3 6. Decide whether Rolle s Theorem can be applied to f(x) = x 5x on the interval [, ]. If Rolle s Theorem can be applied, find value(s), c, in the interval such that f (c) = 0. If Rolle s Theorem cannot be applied, state why.. Find the horizontal asymptotes (if any) for f(x) = ax number constants. b+cx+. Assume that a, b, c, and d are real 8. Let f(x) = x x +1. Find the following (if possible) and sketch the graph. (Hint: f (x) = (1 x ) a. Vertical asymptote(s): b. Horizontal asymptote(s): c. Intercept(s): d. f (x) 9. Let f(x) = x a. Vertical asymptote(s): b. Horizontal asymptote(s): c. Intercept(s): d. f (x) e. Critical number(s): f. Increasing interval(s): g. Decreasing interval(s): h. Relative maxima: i. Relative minima: j. Inflection point(s): k. Concave up: l. Concave down:. Find the following (if possible) and sketch the graph. (Hint: x + f (x) = e. Critical number(s): i. Relative minima: f. Increasing interval(s): j. Inflection point(s): g. Decreasing interval(s): k. Concave up: h. Relative maxima: l. Concave down: 50. Sketch the graph of a function having the following characteristics: f(0) = f(6) = 0 f (x) < 0 if 0 < x < f (x) > 0 if x < 0 or x > f (x) = 0 if x = 0 or x = f ( ) = 0 (x +1) ) 6x ) (x +) Find the following limits. Give exact answers. a. lim x x +6x +5 8 x+x b. lim x 5x 1 x + 5. A dog owner has 00 feet of fencing to use for a rectangular dog run. If he makes two adjacent pens out of the fencing, what dimensions will maximize the area of each pen? 5. Find two numbers whose product is 6 if the sum of the first number and three times the second number is minimized. 5. Calculate iterations of Newton s Method to approximate the real zero of f(x) = x + x +. Use x 1 = 1 as the initial guess and round to four decimal places after each iteration. Stop when your approximations are within of each other. 55. Apply at least three iterations of Newton s Method to f(x) = x x using the initial guess x 1 = 0. Explain why the method fails. 56. Find the values of dy and y for y = x x when x = and x = Find dy for each of the following: a. y = x + 5 x b. y = tan x x +5 c. y = x 1 + x 58. Use differentials to estimate the value of The side of a cube is measured to be.0 inches. a. If the measurement is correct to within inch, use differentials to estimate the propagated error in the volume of the cube. b. Find the percent error. 60. Evaluate the integral x +8. x 61. Evaluate the integral +5x x. 6. Find y = f(x) if f (x) = x +, f (0) =, and f(0) = 1.

4 6. Evaluate the following limit: 1) lim [(i ( n n n ) i=1 6. Use the limit definition to set-up the definite integral representing the area of the shaded region in the figure to the right: y = (1 x) 65. Use the Fundamental Theorem of Calculus to evaluate x x. 66. Determine which of the following is not equal to axf(x). a. a xf(x) b. x af(x) 6. Evaluate the integral 1 x+1. n ] c. axf(x) 9 d. axf(x) + axf(x) e. None of these. 68. Find the average value of f(x) = x on the interval [0, ]. π. π 69. Evaluate csc x 0. Evaluate the integral sec x tan x. 1. Evaluate x+. (x 1). Use the Trapezoidal Rule and Simpson s Rule to approximate the area of the region bounded by the graphs of y = 0, x =, x = 1, and y = 1 using n =. (x ). Find the derivative of y = ln + cos x.. Solve the differential equation ds dt = sec t tan t sec t+5. Answers: x 9 6. x Prove: lim x (x + ) = 16. Proof: Let > 0 and take δ = ε. Suppose 0 < x <. Then 0 < x < x 1 < (x + ) 16 <. Hence, lim x (x + ) = 16. // 11. Continuous over the closed interval [, ]. 1. m 5 1. x + y = 1. y = 6x 15. y = x a) 0 b) 1 c) DNE 1. a) b) d) Nonremovable at x = 1 e) To the right c) Nonremovable discontinuity at x = 5. f(x) is continuous on (, 5) (5, ). 18. f(x) = x, g(x) = x x +, h(x) = x. Then lim x 9 h(x) =, lim x g(x) = 1, and lim x 1 f(x) = 1. So lim x 9 (f g h)(x) = 1.

5 19. f(x) is continuous on any interval (since a polynomial). f(1) = 1 < 0 and f() = 1 > 0. So f(x) has a zero between x = 1 and x =. 0. ( 1, 1 ) and (1, ). s sec t tan t (t) = 1. dy/ = 9 sec t x + 8 sin x + 1x. dy/ = x(x + 1) (x 5)(1x 1x. dy 5) = y 5x y x. x. dy = x +6xy x y x 1 6x y+x. y = 10x 5. dy = 1; y = x + (1 x ) 5. f (x) = x(5x ) 6 cos(sin x) cos x 6. y = sin (sin x). y = 6 tan (x ) sec (x ) 8. f (x+ x) (x) = lim + (x +) x m/s 0. A 1. a) after 8 seconds b) 5 m/s x = x 6. y = 10 (x+) in/sec cm /hr. relative (, ); relative (, ) 9. The MVT applies; c = 0. MVT can be applied; c = 6 1. absolute x = b. C. critical numbers: x = π, 5π ; increasing on (π, 5π ), decreasing on (0, π ) (5π, π) 5. negative 6. Rolle s Theorem applies; c = 5. There are no horizontal asymptotes. 8. a) none b) y = c) (0, 0) d) f (x) = x (x +1) e) x = 0 f) (0, ) g) (, 0) h) none Graph to the right. 9. a) none b) y = and y = c) (0, 0) d) f (x) = e) none f) (, ) g) none 1 (x +), 0, 5 i) (0, 0) j) x = ± k) (, ) l) (, ) (, ) h) none i) none j) (0, 0) k) (, 0) l) (0, ) Graph to the right. 50. Answers may vary. Graph must increase on (, 0) (, ), decrease on (0, ), have a x = 0, have a x =, cross the (0, 0) and (6, 0), and have an inflection x = 51. a) 0 b) ft by 50 ft and

6 5. x 1 = 1 x = 0.65 x 5 = x = 0.51 x = x 1 = 0 x = 1 x = dy = 0.1; y = a) dy = (x 5 ) x x = 1 sequence does not converge b) dy = (x sec x+5 sec x x tan x) (x +5) c) dy = (x +1) 1+x ( 1.) = a).05 in b) 0.6% x + C x x + 5 x + C 6. y = 1 x + x + x n 6. lim n [(1 i n ) i=1 ] n B 6. x C sec x + C Trap 0.8; Simp dy = cos x sin x +cos x. s(t) = ln sec t C

NO CALCULATOR 1. Find the interval or intervals on which the function whose graph is shown is increasing:

NO CALCULATOR 1. Find the interval or intervals on which the function whose graph is shown is increasing: AP Calculus AB PRACTICE MIDTERM EXAM Read each choice carefully and find the best answer. Your midterm exam will be made up of 5 of these questions. I reserve the right to change numbers and answers on