Midterm 1 - DataOverall (all sections):

Average 75.12Median 78.50Std dev 15.40

Section 80:Average 74.77Median 78.00Std dev 14.70

Midterm 2 - DataOverall (all sections):

Average 74.55Median 79Std dev 18.55

Section 80:Average 74.06Median 78.00Std dev 17.68

Real Grades VS Expected Grades (Midterm 1)

Real Grades VS Expected Grades (Midterm 2)

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 thesemistakes.

Álvaro Lozano-Robledo (UConn) MATH 1131Q - Calculus 1 9 / 46

Read This First!Please read each questioncarefully. In order to receivefull credit on a problem, thesolution 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, circlethe printed capital F. In each case, write a careful and clear justificationor 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:

Second Midterm Exam, Problem 1:

(c) The limit limx→0

1− cos(2x)

x2 + xis equal to 1. T F

Justification:

(d) If f (x) = ln(10) for all x , then f ′(x) =1

10. T F

Justification:

(e) To find the absolute maximum and minimum values of a continuousfunction f (x) on a closed interval, it is enough to compare the values atthe end points of the closed interval. T FJustification:

Second Midterm Exam, Problem 2:

On the curve sin(xy) = x2 + y2, computedy

dxin terms of x and y .

Second Midterm Exam, Problem 3: Find the derivatives of thefollowing functions. You do not have to simplify.

(a) y = xcos x .

(b) f (x) = sin(e3x)

Second Midterm Exam, Problem 4: Use calculus to find the absolutemaximum and minimum values of f (x) = 3x4 − 4x3 − 12x2 + 12 on theinterval [−1,3]. Explain how you found these values.

Absolute maximum value: Absolute minimum value:

Second Midterm Exam, Problem 5: A sample of a radioactivesubstance 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 originalamount?

Second Midterm Exam, Problem 6: A paper cup has the shape of aright circular cone with height 12 cm and radius 4 cm (at the top).Water is poured into the cup at a rate of 2 cm3/s. Use calculus todetermine how quickly the water level is rising when the water is 6 cmdeep. (The general formula for the volume of a right circular cone isV = 1

3πr2h.)

h

Second Midterm Exam, Problem 6: A paper cup has the shape of aright circular cone with height 12 cm and radius 4 cm (at the top).Water is poured into the cup at a rate of 2 cm3/s. Use calculus todetermine how quickly the water level is rising when the water is 6 cmdeep. (The general formula for the volume of a right circular cone isV = 1

3πr2h.)

Second Midterm Exam, Problem 7: Suppose the first derivative of a

function f (x) is given by f ′(x) =−8x

(x2 − 4)2. Moreover, it is given to you

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 increasingand the open intervals where f (x) is decreasing. Make it clearwhich is which. Give the endpoints of the intervals exactly, not asdecimal approximations.

Second Midterm Exam, Problem 7: Suppose the first derivative of a

function f (x) is given by f ′(x) =−8x

(x2 − 4)2. Moreover, it is given to you

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) =24x2 + 32

(x2 − 4)3.

Make it clear which is which. Give the endpoints of the intervalsexactly, not as decimal approximations.

Second Midterm Exam, Problem 7: Suppose the first derivative of a

function f (x) is given by f ′(x) =−8x

(x2 − 4)2. Moreover, it is given to you

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).

Second Midterm Exam, Problem 8: A closed box (top, bottom, andall four sides) needs to be constructed to contain 9 m3 and have abase whose width is twice its length. Use calculus to determine thedimensions (length, width, height) of such a box that uses the leastamount of material.

Second Midterm Exam, Problem 9: Use calculus to compute thefollowing limits.(a) lim

x→0+(1− 2x)1/x

Second Midterm Exam, Problem 9: Use calculus to compute thefollowing limits.

(b) limx→0

e3x − x − 1

x

Second Midterm Exam, Problem 10: Find the linearization of thefunction ex−1 at x = 1.