Midterm 1 - Data. Overall (all sections): Average Median Std dev Section 80: Average Median Std dev 14.

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Marilyn Farmer
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2 Midterm 1 - Data Overall (all sections): Average Median Std dev Section 80: Average Median Std dev 14.70

4 Midterm 2 - Data Overall (all sections): Average Median 79 Std dev Section 80: Average Median Std dev 17.68

7 Real Grades VS Expected Grades (Midterm 1)

8 Real Grades VS Expected Grades (Midterm 2)

9 So... the grading of the Second Midterm exam was... (A) You were way too harsh. Take it easy on us! (B) Tough! (C) Fair. (D) Kind of easy grading. (E) So easy... can t believe I got away with these mistakes. Álvaro Lozano-Robledo (UConn) MATH 1131Q - Calculus 1 9 / 46

10 Read This First! Please read each question carefully. In order to receive full credit on a problem, the solution must be complete, nicely organized in the page, logical and understandable.

Second Midterm Exam, Problem 1: If the statement is always true, circle the printed capital T. If the statement is sometimes false, circle the printed capital F.

11 Second Midterm Exam, Problem 1: If the statement is always true, circle the printed capital T. If the statement is sometimes false, circle the printed capital F. In each case, write a careful and clear justification or a counterexample. (a) If f (c) = 0 and f (c) < 0, then there must be a local maximum at x = c. T F Justification: (b) If f (c) = 0, then the graph of y = f (x) has an inflection point at (c, f (c)). T F Justification:

12 Second Midterm Exam, Problem 1: 1 cos(2x) (c) The limit lim x 0 x 2 is equal to 1. T F + x Justification: (d) If f (x) = ln(10) for all x, then f (x) = 1 Justification: 10. T F (e) To find the absolute maximum and minimum values of a continuous function f (x) on a closed interval, it is enough to compare the values at the end points of the closed interval. T F Justification:

13 Second Midterm Exam, Problem 2: On the curve sin(xy) = x 2 + y 2, compute dy in terms of x and y. dx

14 Second Midterm Exam, Problem 3: Find the derivatives of the following functions. You do not have to simplify. (a) y = x cos x. (b) f (x) = sin(e 3x )

15 Second Midterm Exam, Problem 4: Use calculus to find the absolute maximum and minimum values of f (x) = 3x 4 4x 3 12x on the interval [ 1, 3]. Explain how you found these values. Absolute maximum value: Absolute minimum value:

16 Second Midterm Exam, Problem 5: A sample of a radioactive substance decayed to 95 % of its original amount after a year. 1 What is the half-life of the substance? 2 How long would it take the sample to decay to 15 % of its original amount?

17 Second Midterm Exam, Problem 6: A paper cup has the shape of a right circular cone with height 12 cm and radius 4 cm (at the top). Water is poured into the cup at a rate of 2 cm 3 /s. Use calculus to determine how quickly the water level is rising when the water is 6 cm deep. (The general formula for the volume of a right circular cone is V = 1 3 πr 2 h.) h

18 Second Midterm Exam, Problem 6: A paper cup has the shape of a right circular cone with height 12 cm and radius 4 cm (at the top). Water is poured into the cup at a rate of 2 cm 3 /s. Use calculus to determine how quickly the water level is rising when the water is 6 cm deep. (The general formula for the volume of a right circular cone is V = 1 3 πr 2 h.)

19 Second Midterm Exam, Problem 7: Suppose the first derivative of a function f (x) is given by f (x) = 8x. Moreover, it is given to you (x 2 4) 2 that f (0) = 0 and f (x) has vertical asymptotes at x = 2 and at x = 2. (a) Use calculus to find the open intervals where f (x) is increasing and the open intervals where f (x) is decreasing. Make it clear which is which. Give the endpoints of the intervals exactly, not as decimal approximations.

20 Second Midterm Exam, Problem 7: Suppose the first derivative of a function f (x) is given by f (x) = 8x. Moreover, it is given to you (x 2 4) 2 that f (0) = 0 and f (x) has vertical asymptotes at x = 2 and at x = 2. (b) Use calculus to find the open intervals where the graph of y = f (x) is concave up and concave down given that f (x) = 24x (x 2 4) 3. Make it clear which is which. Give the endpoints of the intervals exactly, not as decimal approximations.

21 Second Midterm Exam, Problem 7: Suppose the first derivative of a function f (x) is given by f (x) = 8x. Moreover, it is given to you (x 2 4) 2 that f (0) = 0 and f (x) has vertical asymptotes at x = 2 and at x = 2. (c) Use the information above to sketch the graph of the function f (x).

22 Second Midterm Exam, Problem 8: A closed box (top, bottom, and all four sides) needs to be constructed to contain 9 m 3 and have a base whose width is twice its length. Use calculus to determine the dimensions (length, width, height) of such a box that uses the least amount of material.

23 Second Midterm Exam, Problem 9: Use calculus to compute the following limits. (a) lim x 0 +(1 2x)1/x

24 Second Midterm Exam, Problem 9: Use calculus to compute the following limits. (b) lim x 0 e 3x x 1 x

25 Second Midterm Exam, Problem 10: Find the linearization of the function e x 1 at x = 1.

St. Augustine, De Genesi ad Litteram, Book II, xviii, 37. (1) Note, however, that mathematici was most likely used to refer to astrologers.

St. Augustine, De Genesi ad Litteram, Book II, xviii, 37. (1) Note, however, that mathematici was most likely used to refer to astrologers. Quote: [...] Beware of mathematicians, and all those who make empty prophecies. The danger already exists that the mathematicians (1) have made a covenant with the devil to darken the spirit and to confine