BC -2 Semester Review
This is a beginning. It is not intended as a complete review. It is a reminder of the many types of problems we have done this semester and it is a guide for active study. Remember to review old tests and quizzes, problems from the book, and handouts, as well as chapter review sections. Pace yourself while studying during the coming week or so. Remember topics that were clear in January still deserve some time now. Also, please spend some time thinking about the conceptual meaning behind the problems on the following pages and not just the mechanics of getting through each problem.
Exam: Tuesday, 30 May 2017, 1:00-3:00 PM, in Main Gym
Be sure to bring your textbooks!
Topics: / Problems from the book:Derivative applications
· Interpreting families of functions
· Critical points, local/global maxs/mins
· Optimization
· L'Hôpital's Rule
· Parametric equations – graphs, derivatives, tangents, motion, speed
Integrals
· Definition – limit of a sum, signed area
· Fundamental Theorem of Calculus (both versions)
· Properties
· Riemann sums (left, right, midpoint), trapezoidal approx., calculator approx.
· Antiderivatives (constructed graphically, numerically, analytically)
· Differential equations (basic concept)
Integral Applications
· Distance traveled, average value
· Signed area, area between curves
· Arc length
· Volume (shells, washers, slices)
Integration (techniques)
· Powers, polynomials, exponential, trig, inverse trig, log
· Substitution (including changing limits!)
· Integration by parts
· Partial fractions
· Trig powers
· Trig substitution / ch 4
P. 197–9: 6, 10, 12, 13, 21, 26, 35, 38, 43, 47
P. 207–9: 4, 5, 11, 22, 28, 30, 32, 33, 40, 41
P. 234: 18–21, some from 22–35, 39, 43-47, 53
P. 242-5: 8, 13, 14, 19, 21, 23, 26, 37, 45, 52
ch 5
P. 262–4: 3, 5, 9, 14, 16, 18, 23, 26, 27
P. 269–71: 4, 7, 17, 21, 24, 32, 36
P. 278–281: 7, 16, 21ab, 27, 31, 33, 36, 37, 39
P. 288–9: 2, 8, 21–26, 34, 37, 39, 42, 43
ch 6
P. 303–5: 6, 11, 15, 17, 22, 24
P. 310–1: some from 1–63, 67, 71, 74, 77, 81
P. 315–7: 5, 9, 14, 15, 17, 21, 25
P. 320–1: 3, 11, 22, 27, some from 29–36, 38
P. 324: 4, 5, 7
ch 7: assorted integrals (lots) to practice techniques
P. 338–40: some from 3–40, some from 47–62, some from 69–76, 89, 94, 99, 109
P. 346–7: some from 2–28, 36, 37, 39, 46, 56, 58
P. 351: 9, 11, 12, 45, 46
P. 359–60: some from 8–22, some from 27–58
P. 365–6: 11, 14, 15, 18, 20
P. 385–387: lots from 1–156!
ch 8
P. 396–7: 5–8 (set-up integrals, no Riemann sums)
P. 404–5: some from 1–26, 38–42, 47, 49
More goodies to do – to complement the problems in the book, not replace them:
(1) Consider the function , where a and b are constants.
(a) Find all critical points of f.
(b) For what values of a and b does f have exactly one critical point? What are the coordinates of this one critical point, and is it a local maximum, local minimum, or neither?
(c) For what values of a and b does f have exactly three critical points? What are the coordinates of these critical points? Which are local maxima, which are local minima, and which are neither?
(2) Draw the graph of a continuous function that has three different types of critical points on the interval [–5, 5].(3) Six squares are cut out of a piece of cardboard 24 cm by 16 cm. The sides are folded up and the top is folded around to form a closed box with double walls on 3 sides. Find the value of x to maximize the volume of the box. /
(4) A rectangle of fixed perimeter 36 inches is rotated around one of its sides, sweeping out a figure in the shape of a right circular cylinder. What is the maximum possible volume of that cylinder?
(5) A rectangular plot of land containing 216 square meters of area is to be enclosed by a fence and divided into three equal parts by two other fences that will both run parallel to the same side of the rectangular plot. What dimensions for the entire rectangular plot require the smallest total amount of fencing? How much total fencing will be needed?
(6) A commuter train carries 600 passengers each day from a suburb to a city. The cost to ride the train is $4.50 per person. Market research reveals that 40 fewer people would ride the train for each 25 cent increase in fare and 40 more for each 25 cent decrease. What fare should be charged to generate the largest possible revenue?
(7) Determine each of the following limits.
(a) (b)
(c) (d)
(8) Find for each of the following:
(a) (b)
(9) Consider the curve given by the parametric equations .
(a) Write the equation of the tangent line to this curve when .
(b) If P is a point moving along this parametric curve, find the speed of P when .
(c) Eliminate the parameter θ and write the Cartesian (x-y) equation of the curve.
(10) Consider the cycloid given by the parametric equations and . Determine all values of x (exact) for which the tangent to this curve is:.
(a) horizontal (b) vertical
(11) Find the general solution of each DE:
(a) (b)
(12) Find the solution to each initial value problem:
(a) (b)
(13) Dr. Hasler throws his left tennis shoe directly up with a velocity of 25 ft/sec while sitting (carefully) at the top of entelechy (which is 20 feet tall). Two seconds later he throws his right tennis shoe upward with a velocity of 4 ft/sec.. Assuming that the acceleration due to gravity is
(a) How high does his left shoe go?
(b) Which shoe reaches the ground first?
(14) Given the definite integral , complete the following.
(a) Using n = 4, approximate the value of this integral with a left–hand Riemann sum, a midpoint Riemann sum, and a trapezoidal approximation. Write out these sums clearly.
(b) Which of the approximations used in part (a) overestimates the value of the integral? Which underestimates? Clearly explain your reasoning.
(c) Use your calculator to approximate the integral more accurately to 3 decimal places.
(d) Use sigma notation to express the right-hand Riemann sum for this integral for n = 40.
(15) You and your bff are driving along a twisty dirt road in your old "beater" car. The speedometer on this car works but not the odometer. To determine how long this particular road is, you record the car's speed, s, in feet per second (being conscientious mathematics students, you converted from miles per hour as you recorded the data) at different times, t, in seconds during the two-minute time interval you were on the road. Those results are shown in the table below.
s / 0 / 44 / 35 / 15 / 30 / 44 / 35 / 15 / 22 / 35 / 44 / 30 / 35
Using a Riemann sum with 6 equal subintervals, provide the largest possible upper estimate for the length of this road. Explain why this is the largest upper estimate for all Riemann sums with 6 equal subintervals.
(16) Express as a limit.
(17) The limit can represent an integral. What integral?
(18) Let where f is the function graphed at the right. Note: The graph of f consists of 4 line segments and a quarter-circle, and the domain of f is [–6, 5].(a) Determine F(–5) and F(5).
(b) Determine .
(c) How many zeros does F have on [–6, 5]?
(d) On what intervals of x is F increasing? /
(e) At what values of x does F have a local maximum?
(f) At what values of x does F have inflection points?
(g) On what intervals of x is F concave down?
(19) Find each of the following:
(a) . (b)
(20) Evaluate each of the following integral WITHOUT using the Fundamental Theorem of Calculus!
(a) (b) (c)
(21) The velocity of a particle is given by . Use integrals to find each of the following.
(a) the displacement of the particle for 0 £ t £ 4.
(b) the total distance traveled by the particle for 0 £ t £ 4.
(22) Find the average value of .
(23) Find the (positive) area enclosed by the graphs of:
(a) and the x–axis
(b)
(c)
(24) Find the exact arc length for each of the following curves:
(a) on [0, 2]. (b) on [1, 3].
(25) Set up integrals to find the arc lengths for each of the following curves. Then approximate each length on your calculator, accurate to 3 decimal places:
(a) on [1, 4] (b) on
(26) Find the volume of the solid formed when the region bounded by the graphs of and
x = ln(3) in the first quadrant is revolved around:
(a) the y–axis (b) the x –axis (c) the line y = –2
(27) Find the volume of the solid formed when the region bounded by the graphs of
, y = 4, and x = 3 is revolved around:
(a) the y–axis (b) the x –axis (c) the line x = 4
(28) The base of a solid is a circle with radius a, centered at the origin. Find the volume of the solid if cross–sections perpendicular to the x–axis are:
(a) squares, edge along base (b) semi–circles, diameters on the base
(c) equilateral triangles (d) isosceles right triangles, hypotenuses on the base