Midterm 2
Calculus I (Math 124)
Problem 1.
(a) Find the derivative of f(x) = arctan( √ x). (Recall that arctan(x) is the same function as the inverse tangent function tan−1 (x).)
(b) If y = ln(x) ln(x) , find an expression for dy/dx in terms of x.
Problem 2. Consider the curve defined by the equation x 4 − 2x 2y 2 = −56.
(a) Find the equation of the tangent line at the point (2, 3).
(b) Let P be the point on the curve near (2, 3) with x-coordinate 2.1. Find an approximate value of the y-coordinate of P. (Please round your answer to three digits after the decimal.)
Problem 3. An hourglass is made up of two glass cones connected at their tips (as in the diagram below). Both cones have radius 1 cm and height 2 cm. When the hourglass is flipped over, sand starts falling to the lower cone.
(a) When the sand remaining in the upper cone has height y cm, give a formula for its volume A in terms of y.
(b) When the sand in the lower cone has reached a height of h cm, give a formula for its volume B in terms of h.
(c) Assume that the total volume of sand is 2π/3 cm3 and that the height of the sand in the upper cone is decreasing at a rate of 1 cm/sec. At the instant that the sand in the lower cone is 1 cm high, determine the rate at which the height of the sand in the lower cone is increasing.
Problem 4. Beginning at time t = 0 seconds, an ant crawls according to the equations
x(t) = t3 + 45t + 1 and y(t) = −12t2.
(a) At what times t within the first 10 seconds is the ant’s direction of travel parallel to the line x + y = 2? (Please round your answer to three digits after the decimal.)
(b) At what time within the first 10 seconds does the ant attain its maximal speed? (Please round your answer to three digits after the decimal.)
Problem 5. Consider the function
f(x) = x4 − 6x2 + 4.
(a) Using interval notation, determine where f(x) is increasing and decreasing.
(b) Determine the critical numbers of f(x).
(c) Using interval notation, determine where f(x) is concave up and down.
(d) Determine the inflection points of f(x).
(e) For each critical number, determine whether it is a local minimum, local maximum or neither.