Download Final review

MA1113: Single Variable Calculus Part 1
Name ________________
Take Home Exercise 3
Part A: Knowledge of Terminology and Symbology
____ Newton's Method
A. Any value of x where f "(x)=0 and f '' is changing sign
____ Optimization
B. The rate of change of the first derivative (measures curvature)
____ Local Maximum
C. Name for an infinite sum of infinitesimally small quantities
____ Local Minimum
D. A sloping line approached by a function as x approaches 
____ Critical Number
E. Criterion that a function satisfies wherever it is increasing
____ Integration
F. Criterion that a function satisfies wherever it is decreasing
____ Net Area
G. Criterion corresponding to concave down curvature
____ Ledge
H. Function feature corresponding to f ''(x) > 0 when f '(x)=0
____ f '( x)  0
I. Criterion that corresponds to concave up curvature
____ Inflection Point
J. Identifies integration and differentiation as inverse processes
____ Change of Variable
K. A particular type of inflection point where f '(x) = 0
____ Second Derivative
L. Certain limits that can be evaluated via L'Hospital's Rule
____ f '( x)  0
M. Function feature associated with an f '(x) change from + to -
____ Slant Asymptote
N. A geometric interpretation of any definite integral
____ f "( x)  0
O. Terminology that is equivalent to the "indefinite integral"
____ f "( x)  0
P. Any value of x for which f '(x)=0 or f '(x) is undefined
____ Indeterminate Forms
Q. Using the Derivative Product Rule to evaluate some integrals
____ Antiderivative
R. Process to find an extremum of a real-world-based function
____ Integration by Parts
S. Technique that uses the Chain Rule to evaluate some integrals
____ Fund. Theorem of Calculus
T. Function feature associated with an f '(x) change from - to +
U. Uses repeated linearizations to precisely locate a function's root
V. Function feature corresponding to f "(x)<0 when f '(x)=0
-1-
Part B: Knowledge of Important Formulas
1. L'HOSPITAL'S RULE:
a. Complete the formula for L'Hospital's Rule is:
lim
x a
f ( x)

g ( x)
b. Indicate at right the indeterminate forms to which the rule may
be directly applied (i.e., without any algebraic manipulation):
c. After applying the rule once, how do you
know if the rule must be applied again?
d. Write the proper formula for a second
application of L'Hospital's Rule at right:
e. How do you know when you no longer
need to apply L'Hospital's Rule?
2. RELATIONSHIPS BETWEEN BASIC DIFFERENTIATION AND INTEGRATION FORMULAS
a) If
d n
x  
dx  
then the indefinite integral (antiderivative)

dx 
b) If
d x
e  
dx  
then the indefinite integral (antiderivative)

dx 
c) If
d
ln x  
dx
then the indefinite integral (antiderivative)

dx 
d ) If
d
sin x  
dx
then the indefinite integral (antiderivative)

dx 
e) If
d
cos x  
dx
then the indefinite integral (antiderivative)

dx 
f ) If
d
 tan x  
dx
then the indefinite integral (antiderivative)

dx 
g ) If
d
sin 1 x  
dx
then the indefinite integral (antiderivative)

dx 
h) If
d
cos 1 x  
dx
then the indefinite integral (antiderivative)

dx 
h) If
d
 tan 1 x  
dx
then the indefinite integral (antiderivative)

dx 
-2-
3. INTEGRATION RULES
a) If F(x) and G(x) are the antiderivatives of f(x) and
g(x) and k is a constant, complete the following rules:
k
k

dx 
d
(uv) 
b) Complete the Product Rule:
dx
and show how it can be turned
into the Integration By Parts Rule:
f ( x) dx 
f ( x)  g ( x)
 dx 
Part C: Concept Visualization
f ( x)
y
1. NEWTON'S METHOD
Derive the basic formula used in Newton's
Method to find the root x=r of f(x) using the
following outline (which will NOT be supplied
on the final exam!)
xr
a) Sketch in the linearization L1(x) of f (x) that
is tangent to f(x) at x = x1, and write below the
formula for the slope of L1:
x1
m1 =
b) Show how you can
derive the formula for L1(x)
using the point-slope
formula for a straight line.
c) Indicate on the graph the
location of the root of L1(x),
label it x2, and derive the
formula for this root.
c) On the sketch indicate how the next linearization
root x3 is found and simply write the formula at right:
x3 
xn1 
d) At right simply write down the formula for the
linearization root xn+1:
-3-
x
2. RIEMANN SUMS
a. For the hypothetical function shown in
the figures at right, approximate the integral:
4
(a)
5
 f ( x)dx
3
0
2
using a 5-term Riemann Sum with
* left-hand sampling points in figure (a),
1
* right-hand sampling points in figure (b).
In each case, shade in the area that each Riemann Sum represents.
1
4
2
3
4
5
2
3
4
5
(b)
3
2
1
1
3. INTEGRATION AS AN AREA COMPUTATION
a) For each integral below: i) make a sketch of the integrand, ii) shade in the total area represented by the
integral, iii) identify which portions of the total area are positive or negative, iv) give the value of each
integral based on your sketches.

2
x
 sin x dx  ___ 
3
dx  ___ 
2

b) A general rule based on the above illustrations is: "The integral of any function with ______ symmetry
over limits that are ___________ with respect to the y-axis is always identically ________."
c) For each integral below: i) make a sketch of the integrand, ii) shade in the total area represented by the
integral, iii) identify which portions of the total area are positive or negative, iv) give an equivalent integral
that is easier to evaluate
 /2
2
2
(
4

x
) dx 

 cos x dx 
 / 2
2
-4-
Part D: Problem Solving
1. Evaluate the following limits using L'Hospital's Rule after suitable algebraic manipulation
1
a)
b) lim ( x  1) 2 x
lim x ln x
x 0
x 0
2. Use Newton's Method to obtain an estimate for:
x3 6
beginning with a 1st guess of x1=2, and proceeding as follows:
a) Show how the above problem can be transformed into
finding the root of a polynomial. Provide a rough sketch of
the polynomial at right and indicate the location of the root.
b) The formula on which Newton's Method is based is given at right.
Use this formula as many times as necessary to find the root estimate x3.
Perform all your calculations using 5 decimal places.
-5-
xn 1  xn 
f ( x)
f ' ( x)
Function Shape Analysis Worksheet
Problem: D-3: f ( x)  y  x3  6 x 2  9 x
1. Find y-intercept and x-intercepts
(if readily evident– e.g., f(x) factorable)
y-intercept: y = _____
x-intercepts (roots): x = ________________
2. Determine F(x) Behavior at Infinity
(i.e., find Asymptotes, if any exist)
Horizontal: y = _________ Slant: y = ___ x
Vertical:
x = _________
3. Find Critical Numbers x= c such that f ' (c)  0
Critical Number Summary
f ' = 0 at x = __________
f ' DNE at x = ___________
4. Do 1st Derivative Trend Analysis at Critical Point f '(c)=0 to Identify Local Maxs / Mins / Ledges
f ' trend around x = ____:
If x  ____(
), then
y'  (
)  (
)  (
)
If x  ____(
), then
y'  (
)  (
)  (
)
f ' trend is ___ to ___ = ________
f ' trend around x = ____:
If x  ____(
), then
y'  (
)  (
)  (
)
If x  ____(
), then
y'  (
)  (
)  (
)
f ' trend is ___ to ___ = ________
f ' trend around x = ____:
If x  ____(
), then
y'  (
)  (
)  (
)
If x  ____(
), then
y'  (
)  (
)  (
)
Critical No. Coords: ( x, y) = _______________
f ' trend is ___ to ___ = ________
(x,y) = _______________
(x,y) = _______________
Increasing Intervals:
___________________________________________________________
Decreasing Intervals:
___________________________________________________________
BASED ON 1st DERIVATIVE TREND ANALYSIS:
At x = ___ , f(x) has a local (max) / (min) / (ledge)
At x = ___, f(x) has a local (max) / (min) / (ledge)
At x = ___ , f(x) has a local (max) / (min) / (ledge)
At x = ___, f(x) has a local (max) / (min) / (ledge)
-1-
Function Shape Analysis Worksheet (cont)
Problem: _____________
5. Find Inflection Points x= c such that
f(x) =
f'(x)=
f " (c)  0 and identify intervals of
f "( x)  0 and
f "( x)  0
Infection Point Summary
f '' = 0 at x = __________
f '' DNE at x = ___________
Concavity Analysis:
If _____ < x < _____, then y ''  (
)  (
)  (
)
i.e., f(x) concave ___ on this interval
If _____ < x < _____, then y ''  (
)  (
)  (
)
i.e., f(x) concave ___ on this interval
If _____ < x < _____, then y ''  (
)  (
)  (
)
i.e., f(x) concave ___ on this interval
If _____ < x < _____, then y ''  (
)  (
)  (
)
i.e., f(x) concave ___ on this interval
Inflection Coords: ( x, y) = _______________ (x,y) = ________________ (x,y) = ________________
Concave UP Intervals: ______________________________________
Concave DN Intervals: _____________________________________
6. Do 2nd Derivative Sign Analysis
at each Critical Point: f '(c) = 0
Local Maximums
where f "(c)  0
to identify
and
Local Minimums
where f "(c)  0
Note 1: If f "(c) = 0 then the 2nd Derivative Sign Analysis is indeterminate. (provides no information)
Note 2: Your results here should be consistent your results from the 1st Derivative Trend Analysis on page 1
When f '(
)  0,
f ''(
)  (
)  (
)  (
)
i.e., concave ____ which is a ______
When f '(
)  0,
f ''(
)  (
)  (
)  (
)
i.e., concave ____ which is a ______
When f '(
)  0,
f ''(
)  (
)  (
)  (
)
i.e., concave ____ which is a ______
BASED ON 2nd DERIVATIVE SIGN ANALYSIS:
At x = ___ , f(x) has a local (max) / (min) / (ledge)
At x = ___, f(x) has a local (max) / (min) / (ledge)
At x = ___ , f(x) has a local (max) / (min) / (ledge)
At x = ___, f(x) has a local (max) / (min) / (ledge)
-2-
Function Shape Analysis Results
On your sketch of the function be
sure to:
4
* draw a symbol at the location
of any roots, local maxs or mins,
and inflection points
3
2
* Label the above points with
MAX, MIN, or IP as appropriate
1
* identify the intervals over which
the function's curvature is concave
up (CU) or concave down (CD)
-4
-3
-2
-1
1
2
3
4
-1
-2
-3
-4
4. A cylindrical drum is required to have a volume of 32π cubic feet. If the material
costs of the drum are $2 per square foot for the top and bottom, and $1 per square foot
for the side, find the dimensions r and h that will result in the minimum material cost
for the drum?
-8-
r
h
5. Use the Integration Sum Rule to evaluate:

3
6
5
3
  5 sin x  x  2 cosh x  4 x  1  x 2


dx


3
 (4  x
6. With regard to the integral:
2
)dx
3
a) Make a sketch of the integrand from x = -3 to +3.
b) Identify on the sketch the net area represented by the integral.
c) Evaluate the integral to find the value of the net area.

7. With regard to the integral:
 (1  cos x)dx
2

a) Make a sketch of the integrand from x = - to .
b) Identify on the sketch the net area represented by the integral.
c) Evaluate the integral to find the value of the net area.
+
-
-2
-9-
8. Sketch the integrand from [-π/2 , π/2 ], shade the
area being found by the integral and then use Change
of Variable technique to estimate that area to three
significant digits.
 /2

x cos( x 2 )dx
1
0
+/2
-/2
-1
1
9. Use the Substitution Technique (Change of Variable) to evaluate:
2x
dx
2
1
 3x
0
10. For the definite integral at right:
a) Sketch the integrand from [-π , π ] and shade the area being found.
b) Then use the Substitution Technique (Change of Variable) and
formula 65 from your Reference Cards to evaluate the integral.
 /2
 tan
0
2
 x
  dx
 2
1
+
-
-1
-10-
11. Use Integration by Parts to evaluate:

x 2 ln( x) dx 
 /2
12. For the definite integral at right:
a) Sketch the integrand and the area being found on the graph.
b) Then use Integration By Parts to evaluate the integral.
 (2 x cos x)dx
0
2
+/2
-/2
-2
 / 2
13. If the integral in problem 3 above had been:
(Note the different lower integration limit!)

(2 x cos x) dx
2
 / 2
a) Sketch the integrand and the area being found on the graph.
b) Determine the value of the integral using the simplest
method possible. Explain your reasoning.
+/2
-/2
-2
-11-