Math 206 Practice Test 3

Transcription

1 Class: Date: Math 06 Practice Test. The function f (x) = x x + 6 satisfies the hypotheses of the Mean Value Theorem on the interval [ 9, 5]. Find all values of c that satisfy the conclusion of the theorem. a. 8, 6 b. 8, 7 c. 7 d. 6 x 7. The function f (x) = e satisfies the hypotheses of the Mean Value Theorem on the interval [0, 4]. Find all values of c that satisfy the conclusion of the theorem. ˆ a. 7ln e e ˆ b. 7 ln e 7e ˆ c. 7ln e 7e ˆ d. 7 ln e e

2 . Verify that the function satisfies the three hypotheses of Rolle's Theorem on the given interval. Then find all numbers c that satisfy the conclusion of Rolle's Theorem. f( x) = x 5x + 6x +, [ 0, 4] a. c = b. c = + c. c = +, c = d. c = 5 + 7, c = 5 7 e. c = +, c = 4. Given f (x) = x + 6x. (a) Find the intervals on which f is increasing or decreasing. (b) Find the relative maxima and relative minima of f. a. (a) Increasing on (, ), decreasing on (, ) (b) Rel. min. f ( ) = 9 b. (a) Increasing on (, ), decreasing on (, ) (b) Rel. max. f ( ) = 9 c. (a) Increasing on ( 6, ), decreasing on (, 6) (b) Rel. min. f ( 6) = 0 d. (a) Increasing on (, 6), decreasing on ( 6, ) (b) Rel. max. f ( 6) = 0

3 5. Given f (x) = x x 6x + 6. (a) Find the intervals on which f is increasing or decreasing. (b) Find the relative maxima and relative minima of f. a. (a) Increasing on (, ) and (, ), decreasing on (, ) (b) Rel. max. f ( ) = 50, rel. min. f () = 75 b. (a) Increasing on (, ) and (, ), decreasing on (, ) (b) Rel. max. f ( ) =, rel. min. f () = 6 c. (a) Increasing on (, ), decreasing on (, ) and (, ) (b) Rel. max. f () = 6, rel. min. f ( ) = d. (a) Increasing on (, ), decreasing on (, ) and (, ) (b) Rel. max. f () = 75, rel. min. f ( ) = Given f (x) = x + x. (a) Find the intervals on which f is increasing or decreasing. (b) Find the relative maxima and relative minima of f. a. (a) Increasing on(, 0) and ( 0, ), decreasing on (, ) and (, ) (b) Rel. max. f () =, rel. min. f ( ) = b. (a) Increasing on (, ) and (, ), decreasing on (, 0) and ( 0, ) (b) Rel. max. f ( ) =, rel. min. f () = c. (a) Increasing on (, ) and (, ), decreasing on (, ) (b) Rel. max. f ( ) =, rel. min. f () = d. (a) Increasing on(, ), decreasing on (, ) and (, ) (b) Rel. max. f () =, rel. min. f ( ) =

4 7. Evaluate the limit. ˆ lim x x + x a. b. c. 0 d. 8. Evaluate the limit lim x 6 ln(x 5) x 6 ˆ using l Hôpital s Rule. a. 0 b. c. d Evaluate the limit. lim x 5 + x x 5 5 lnx ˆ a. b. c. d. 0 e. 4

5 0. Find f. f ( x) = 8x + 4x a. f( x) = 6x + 4x 4 + Cx + D b. f( x) = x + x 4 + Cx + D c. f( x) = 4x + 8x 4 + Cx + D d. f( x) = x + x 4 + Cx + D e. f( x) = x + 4x 4 + Cx + D. At 4:00 P.M. a car's speedometer reads 5 mi/h. At 4:5 it reads 7 mi/h. At some time between 4:00 and 4:5 the acceleration is exactly x mi/h. Find x.. Find the value of the limit. lim x x x. Use l'hospital's Rule to calculate the exact value of the limit f( x) g( x) f( x) = e x 4 and g( x) = x 5 + 6x 4. Find the limit. as x 0. lim y 5y + ˆ y + y y 5. Find the limit. lim x 0 x tan ( x) 6. For what values of a and b is,.5 ( ) is an inflection point of the curve x y + ax + by = 0? What additional inflection points does the curve have? 5

6 7. Find f. f ( x) = x + sinhx, f( 0) =, f( ) = 5 8. Find the most general antiderivative of the function. f( x) = 8x 7 x 0 9. Sketch the graph of f (x) = x on (, ] and find its absolute maximum and absolute minimum values, if any. 0. Sketch the graph of f (x) = sint on 0, π minimum values, if any.. Sketch the graph of the function f (x) = e minimum values, if any. x ˆ and find its absolute maximum and absolute on (, ] and find its absolute maximum and absolute. Find the absolute maximum and absolute minimum values, if any, of the function f (x) = xe 4x on [, ].. Find the absolute maximum and absolute minimum values, if any, of the function f (x) = x lnx x È on, 7 ÎÍ. 6

7 4. You are given the graph of a function f. (a) Determine the intervals on which f is increasing or decreasing. (b) Find the relative maxima and relative minima of f, if any. 5. Given f (x) = 0x x + 6. (a) Find the intervals on which f is increasing or decreasing. (b) Find the relative maxima and relative minima of f. 6. Consider the function f (x) = x e 9x. (a) Find the intervals on which f is increasing or decreasing. (b) Find the relative maxima and relative minima of f. 7. Consider the function f (x) = 8x. (a) Find the intervals on which f is increasing or decreasing. (b) lnx Find the relative maxima and relative minima of f. 8. Determine where the graph of the function u f (u) = u 6 is concave upward and where it is concave downward. Also, find all inflection points of the function. 9. Determine where the graph of the function f (x) = sin9x is concave upward and where it is concave downward on the interval 0 x π 9. Also, find all inflection points of the function. 0. Determine where the graph of f (x) = x e 4x is concave upward and where it is concave downward. Also, find all inflection points of the function. 7

8 . Use the graph of f to find the given limits. (a) lim f (x) x (b) lim f (x) x. Find the limit. sin(7x) lim x 7x. A city s population (in thousands) t years from now is estimated by P(t) = 6t + 0t + 5 t + 5t + 4. What will the population of the city be in the long run? Hint: Find lim P (t). t 4. Find the position function of a particle moving along a coordinate line that satisfies the given condition. v (t) = t 4t + 4, s() = 8

9 Math 06 Practice Test Answer Section. C. A. D 4. A 5. A 6. B 7. C 8. C 9. C 0. B. x = a = 4 b = ( 0,0), (,.5) 7. f( x) = 0 x 5 + sinhx ˆ sinh x x 8 7 0x 0 + C 9. Abs. min. f (0) =

10 0. Abs. max. f π ˆ =. absolute maximum value: f () = e, absolute minimum value: none ˆ. absolute maximum: f 4 =, absolute minimum: f 4e ( ) = e4. absolute maximum: f( 7) = 7ln7 7, absolute minimum: f( ) = 4. (a) Increasing on (, ), decreasing on (, ) and (, ) (b) Rel. max. f () =, rel. min. f ( ) = 5. (a) Increasing on ( 6, 6), decreasing on (, 6) and ( 6, ) (b) Rel. max. f (6) = 5 6, rel. min. f ( 6) = 5 6

11 ˆ 6. (a) decreasing on (, 0) and 9,, increasing on 0, ˆ 9 (b) relative minimum: f (0) = 0, ; ˆ relative maximum: f 9 = 4 8e 7. (a) decreasing on (0, ) and (, e), increasing on ( e, ); (b) relative minimum: f (e) = 8e, relative maximum: none 8. CU on ( 4, 0) and ( 4, ), CD on (, 4) and ( 0, 4), IP ( 0, 0) π 9. CU on 9, π ˆ 9, CD on 0, π ˆ 9, IP π ˆ 9, 0 0. concave upward:, ˆ ˆ 4 and +, 4, concave downward:, + ˆ 4 4, inflection points:, ˆ e + 4 8, +, + ˆ e 4 8. (a) (b). 0. 6, t t + 4t 4