MA1113: Single Variable Calculus Part 1 Name \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ Take Home Exercise 3. Part A: Knowledge of Terminology and Symbology. \_\_\_\_ Newton's Method \_\_\_\_ Optimization \_\_\_\_ Local Maximum \_\_\_\_ Local Minimum
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Name ________________
Take Home Exercise 3
Part A: Knowledge of Terminology and Symbology
____ Newton's Method
____ Optimization
____ Local Maximum
____ Local Minimum
____ Critical Number
____ Integration
____ Net Area
____ Ledge
____
____ Inflection Point
____ Change of Variable
____ Second Derivative
____
____ Slant Asymptote
____
____
____ Indeterminate Forms
____ Antiderivative
____ Integration by Parts
____ Fund. Theorem of Calculus
A. Any value of x where f "(x)=0 andf '' is changing sign
B. The rate of change of the first derivative (measures curvature)
C. Name for an infinite sum of infinitesimally small quantities
D. A sloping line approached by a function as x approaches
E. Criterion that a function satisfies wherever it is increasing
F. Criterion that a function satisfies wherever it is decreasing
G. Criterion corresponding to concave down curvature
H. Function feature corresponding to f ''(x) > 0 when f '(x)=0
I. Criterion that corresponds to concave up curvature
J. Identifies integration and differentiation as inverse processes
K. A particular type of inflection point where f '(x) = 0
L. Certain limits that can be evaluated via L'Hospital's Rule
M. Function feature associated with an f '(x) change from + to 
N. A geometric interpretation of any definite integral
O. Terminology that is equivalent to the "indefinite integral"
P. Any value of x for which f '(x)=0 or f '(x) is undefined
Q. Using the Derivative Product Rule to evaluate some integrals
R. Process to find an extremum of a realworldbased function
S. Technique that uses the Chain Rule to evaluate some integrals
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
1. L'HOSPITAL'S RULE:
a. Complete the formula for L'Hospital's Rule is:
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
2
b) Complete the Product Rule:
and show how it can be turned
into the Integration By Parts Rule:
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:
Part C: Concept Visualization
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!)
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:
m1 =
x
b) Show how you can derive the formula for L1(x) using the pointslope 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:
d) At right simply write down the formula for the linearization root xn+1:
3
(a)
4
a. For the hypothetical function shown in the figures at right, approximate the integral:
3
using a 5term Riemann Sum with
* lefthand sampling points in figure (a),
* righthand sampling points in figure (b).
In each case, shade in the area that each Riemann Sum represents.
2
1
5
1
2
3
4
(b)
4
3
2
1
1
2
3
4
5
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.
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 yaxis 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
4
1. Evaluate the following limits using L'Hospital's Rule after suitable algebraic manipulation
2. Use Newton's Method to obtain an estimate for:
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
Problem: D3:
1. Find yintercept and xintercepts
(if readily evident– e.g., f(x) factorable)
yintercept: y = _____ xintercepts (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
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 = ____:
f ' trend is ___ to ___ = ________
f ' trend around x = ____:
f ' trend is ___ to ___ = ________
f ' trend around x = ____:
f ' trend is ___ to ___ = ________
Critical No. Coords: ( x, y) = _______________ (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
Problem: _____________ f(x) = f'(x)=
5. Find Inflection Points x= c such that and identify intervals of and
Infection Point Summary
f '' = 0 at x = __________
f '' DNE at x = ___________
Concavity Analysis:
i.e., f(x) concave ___ on this interval
i.e., f(x) concave ___ on this interval
i.e., f(x) concave ___ on this interval
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
Local Minimums
where
to identify
and
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
i.e., concave ____ which is a ______
i.e., concave ____ which is a ______
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
On your sketch of the function be
sure to:
* draw a symbol at the location
of any roots, local maxs or mins,
and inflection points
* Label the above points with
MAX, MIN, or IP as appropriate
* identify the intervals over which
the function's curvature is concave
up (CU) or concave down (CD)
4
3
2
1
4 3 2 1
1 2 3 4
1
2
3
4
r
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?
h
8
6. With regard to the integral:
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:
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
1
1
+/2
+
/2

1
1
9. Use the Substitution Technique (Change of Variable) to evaluate:
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.
10
2
+/2
+/2
/2
/2
2
2
11. Use Integration by Parts to evaluate:
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.
13. If the integral in problem 3 above had been:
(Note the different lower integration limit!)
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.
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