MAT 145 Test #4: 100 points

Transcription

1 MAT 145 Test #4: 100 points Name Calculator Used Score Each statement (1) through (4) is FALSE, meaning that it is not always true. For each false statement, either (i) provide a counterexample that disproves the statement (shows when it is false, such as a labeled sketch or an equation with some explanation) or (ii) explain why the statement is false. Be sure to consider extreme cases. 1. If f '(c) = 0, then the function f has a local maximum or local minimum at x = c. (6 pts.) 2. If f ''(3) = 0, then (3, f (3)) is an inflection point of the curve y = f(x). (6 pts.) 3. If a function f has an absolute minimum value at x = c, then f '(c) = 0. (6 pts.) 4. If f '(x) < 0 for 1 < x < 6, then f is increasing on (1,6). (6 pts.) For each question (5) through (8), you are given a situation followed by four statements. Assess each of the four statements and select each statement that is ALWAYS TRUE for the given situation. Each of the four statements is independent of the others. You may select up to four statements for each situation. (32 pts: 2 pts each statement) 5. Suppose f ''(x) < 0 for 4 < x < 4. ttff The function f has no inflection points on ( 4,4). The function s derivative, f ', is decreasing on ( 4,4). If f '(c) = 0 for 4 < c < 4, then f does not need to have a maximum at x = c. The function f is increasing on ( 4,4). 7. Suppose f ' is increasing on ( 1,0) and f ' is decreasing on (0,1). fttf The function f is decreasing on (0,1). The function f is concave up on ( 1,0). The value x = 0 is an inflection point of f. The value x = 0 is a critical point of f. 6. Suppose f is defined for all x on [0,1] and that f ' exists for all x on (0,1). fftt If f is decreasing on (0,1), then f '(x) > 0 on (0,1). If f ''(x) > 0 for all x on (0,1), then f is increasing. If f '(x) > 0 for all x on (0,1), then f has an absolute minimum at x = 0. If f (0) = 0, then f '(c) = f (1) for some value c, 0 < c < Suppose f ' is continuous on [2,6] and that f '(x) < 0 for x < 3 and f '(x) > 0 for x > 3. tfft The function f has a maximum on [2,6]. The function f is increasing on (2,3). The function f is concave upward on (2,6). The function f has a critical point at x = 3.

2 For questions (9) through (11), use the information given to answer each question. Show calculus evidence for your responses and write a sentence to describe the calculus connections between your evidence and your final answer. 9. A function f has derivative f! increasing. (6 pts.) ( x) = x 1 ( x + 2) 2. State all intervals over which the function f is Increasing on: (intervals: exact values) 10. A function f has second derivative f!! of f. (6 pts.) ( x) = 3x x + 4 (x 3) 3. State the location for each point of inflection Inflection Points at: (exact x values) 11. A function f has first derivative f!( x) = 3 x2 + x 2 and second derivative f!! ( x) = 3 x2 + 2x 6. x 3 x 4 Determine the locations of all local maximum and minimum values of f. (8 pts.) Local Mins at: (exact values of x) Local Maxs at: (exact values of x)

3 12. Use the information here to sketch the graph of a function f that meets ALL the following requirements. Label your graph to help me identify these requirements. Then fill in the requested information in the chart. (24 pts: 18 pts evidence to graph; 6 pts info in table) lim f (x) =1 lim f (x) = 2 x x lim f (x) = lim f (x) = x 2 + x 2 There is a local minimum at x = 1 There is an absolute minimum at x = 2 f "(x) < 0 on (, 4 ) $ 1, 1 ' # & % 2 ) f!! (x) > 0 on ( 4, 1) % 1 ( $ 2, 2 & ( ( 2, ) ' f "(x) < 0 on (, 2) ( 0,1) (2, ) f "(x) > 0 on ( 2, 0) ( 1, 2) The y-intercept is (0, 2) The only x-intercept is ( 2,0) A point on the graph is ( 4, 1) A point on the graph is (5, 2) State all intervals over which the function is: Increasing: State the domain and range of the function: Domain: Concave Down: Range:

4 Bonus! Determine the point on the xy-plane, expressed as an ordered pair, indicating where the curves y = x 3 3x + 4 and y = 3( x 2 x) are tangent to each other, that is, have a common tangent line. Include an equation for that common tangent line in the form y = mx + b. Illustrate your result by sketching both curves and their common tangent on the same plane. In your sketch clearly identify each function and the common tangent line. Show all necessary evidence to completely justify your solution. (10 pts) BONUS! BONUS! Given the same two functions as described above, there is only one x-value in the domain of the functions at which the two functions have unique parallel tangent lines. State the x-value at which this phenomenon occurs and generate equations for the two parallel tangent lines. Show all necessary evidence to completely justify your solution. (5 pts)

5 Calculus I MAT 145 Test #4: 100 points Evaluation Criteria You may use a calculator for any questions on the test. (1) thru (4): 24 pts: 6 pts each for complete explanation or appropriate counterexample; responses must meet requirements described with the statement of the question. (5) thru (8): 32 pts (2 pts for each correctly assessed statement) (9) 6 pts: 2 pts correct interval; 4 pts appropriate, clear, and accurate evidence and connections (10) 6 pts: 2 pts correct points of inflection; 4 pts appropriate, clear, and accurate evidence and connections (11) 8 pts: 4 pts correct max/min locations; 4 pts appropriate, clear, and accurate evidence and connections (12) 24 pts (18 pts: each requirement/criteria must be met, your graph must be a function and be accurately presented; 6 pts: intervals, domain/range in chart) BONUS! 15 pts: See question for criteria. Question (12) and any BONUS! responses are to be completed outside the classroom. You may confer with any other students in MAT 145 Section 08. You may use any print or electronic resources. Be sure to cite your references! You may ask me questions. You may not confer with any other people about the BONUS problems in any way shape, or form. Please read the following statement and sign below if this is a true statement for you. I pledge my honor that during both the in-class and take-home portions of this examination I neither gave nor received assistance, and that I saw no dishonest work. (signature) (date)