Chapter 2 Review
I.Evaluate each limit or indicate that it does not exist.
1.2.
3.4.
5.6.
7.8.
9.10.
11.12.
13.14.
15.
II.The function satisfies the inequality
. Find .
III.Let . Find the lim f(x) as
1.2.
3.4.
IV.Find the average rate of change of the function
over the interval .
V.What is the average rate of change of the function g(t) over the
interval from to ? How is it related to the secant line?
VI.Does the existence and value of the limit of a function f(x) as x
approaches c ever depend on what happens at c? Explain, and
give examples.
VII.The graph of a function is shown below. Indicate whether the statement is true or false.
1. exists2.
3.4.
5.
6. exists at every point c in (-1,1).
VIII.Explain why does not exist.
IX.A function g(x) is defined for all real values of x except x = c.
Is it possible to make a conclusion about the?
X.A car is accelerating from a standstill. The graph is shown
below.
1.Estimate the slopes of secant lines PQ1, PQ2, PQ3, and
PQ4.
2.Estimate the speed of the car at time t = 20 sec.
XI.Given that for values of x close to zero.
1.Is there a?
2.Graph , , and together for . Describe the behavior of the graphs as x →0.
XII.For the function whose graph is shown below, state the
following.
1.2.
3.4.
XIII.Determine whether the function is continuous at the given
number. Justifyyour answer.
1.at x = 3
2.at x = -5
XIV.Prove that the function has a zero in the
interval .
XV.Determine the constants a and b so that the following function
is continuous.
XVI.Define in a way that extends to be
continuous at .
XVII.What can be said about the continuity of polynomial functions? trigonometric functions? rational functions?
XVIII.The profits for a small company for each of the first five years
of its operation are given in the following table:
Year / Profit in $1000s2005 / 6
2006 / 27
2007 / 62
2008 / 111
2009 / 174
1.Plot points representing the profit as a function of the
year, and join them by as smooth a curve as you can.
2.What is the average rate of increase of the profits
between 2007 and 2009?
3.Find the average rate of increase of the profits for
and . Average those two rates to get an estimate for the rate at which profits are increasing in 2007.
XIX.From the graph of shown below, state the numbers at which the function is discontinuous. For each point of discontinuity, indicate whether the function is continuous from the right, from the left, or neither.
XX.Find the numbers at which is
discontinuous. At which of these numbers is f continuous from the right, from the left, or neither? Sketch the graph of f.
XXI.If , find the average rate of change of f on the interval .
XXII.Evaluate each limit, indicate that it is ±∞, or that it does not exist, whichever is the best answer.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
XXIII.Sketch a graph for a function that satisfies the given conditions.
, , , , ,
XXIV.For the function g whose graph is given below, state the
following.
1.2.
3.4.
5.
6.The equations of all asymptotes
XXV.Give the equations for the vertical and horizontal asymptotes for each function.
1.
2.
XXVI. Find an equation for the tangent to the curve at the point . Sketch the curve and the tangent together.
XXVII. Find the slope of the tangent line to the graph of the parabola at the point .
XXVIII. Find the indicated limit.
1.
2.
3.
4.
5.
6.
7.
8.
9.
ANSWERS:
I.1.2.
3.4.
5.146.∞
7.8.-∞
9.DNE10.DNE
11.DNE12.0
13.014.
15.0
II.
III.1.2.-∞
3.-14.DNE
IV.0
V.. The average velocity of the function is the slope
of the secant line joining the points and .
VI.No. The function does not have to be defined at x = c to have a
limit as x approaches c. Example: has a limit of 4 as x approaches 2 but isundefined at 2.
VII.1.T2.T
3.F4.F
5.F6.T
VIII., . Therefore, does not exist.
IX.No
X.1.430/10 = 43 m/s; 280/6.5 ≈ 43.077 m/s; 170/3.5 ≈ 48.571
m/s; 100/2 = 50 m/s
2.50 m/s
XI.1.
2.As x→0, the graph of is squeezed between
the graphs of and .
XII.1.-∞2.∞
3.-∞4.∞
XIII.1., , . does not exist or equal . Therefore, f is not continuous at 3.
2., . . Therefore, f is continuous at x = -5.
XIV., . Therefore, for some x
between 0 and 1by the Intermediate Value Theorem.
XV.a = -1, b = 1
XVI.Define to equal 4.
XVII.Polynomial functions are always continuous at every point.
Sine and cosine functions are always continuous. Tangent, cotangent, secant, and cosecant are discontinuous at the asymptotes. Rational functions are discontinuous at any point that makes the denominator zero.
XVIII.2.$56,000/yr
3.Average rate for = $35,000/yr
Average rate for = $49,000/yr
Rate for 2007 = $42,000/yr
XIX.Discontinuous at x = -2, 2, 4, 6, and 8. Continuous from the
right at x = -4and x = 2.
XX.
Discontinuous at x = 0 and x = 1; continuous from the left at x = 0; continuous from the right at x = 1
XXI.8
XXII.1.2.
3.04.1
5.6.
7.08.
9.10.0
11.12.DNE
13.0
XXIII.
XXIV.1.22.-2
3.4.
5.
6.Vert Asy: x = -2 and x = 3
Horiz Asy: y = -2 and y = 2
XXV.1.Vert Asy: and
Horiz Asy:
2.Vert Asy:
Horiz Asy:
XXVI.
XXVII.-4
XXVIII.1.12.3
3.4.1
5.16.2
7.8.
9.