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Strategies for Determining Domains, Ranges, and Graphs of Functions in MTH 261 Final Exam, Exams of Analytical Geometry and Calculus

Missouri State University (MSU)Analytical Geometry and Calculus

Helpful concepts for preparing for the final exam of a mathematics course (mth 261), focusing on strategies for determining the domain, range, and graphical representation of various functions. Topics covered include even and odd functions, graphing piecewise-defined functions, recognizing functions obtained from known functions through vertical and horizontal shifts, and understanding the basic types of graphs for exponential and logarithmic functions. Additionally, the document discusses the definitions and properties of trigonometric functions, their inverse functions, and the computation of limits.

Typology: Exams

Pre 2010

Uploaded on 07/23/2009

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Helpful Concepts for MTH 261 Final

• What are the general strategies for determining the domain of a function?

• How do we use the graph of a function to determine its range?

• How many graphs of basic functions have you already remembered without having to plot

points?

• How do we recognize an even function? What’s special about an even function in terms of

its graph?

• How do we recognize an odd function? What’s special about an odd function in terms of its

graph?

• What is the general procedure in graphing a piecewise-defined function?

• How do we recognize functions whose graphs can be obtained from the graph of a known

function via vertical/horizontal shifts?

• What are the two basic types of graphs for exponential functions?

• What are the basic exponential rules?

• How do we set up an exponential function if its doubling time (or half-life) is known?

• How are the graphs of two functions related if they are inverses?

• What are the two basic types of graphs for logarithmic functions?

• What are the basic logarithmic rules?

• How do we use a logarithm to solve an exponential equation?

• What are the definitions of the six trigonometric functions?

• How many of the trigonometric values of special angles have you already remembered

without having to use a graphing calculator?

• What are the three fundamental trigonometric identities?

• How are the inverse trigonometric functions defined?

• What is the range of each inverse trigonometric function?

• Does the existence of the limit )(lim xf

cx →depend on whether the function f(x) is defined at

x = c? Does it depend on the value f(c) is the function f(x) is defined at x = c?

Partial preview of the text

Download Strategies for Determining Domains, Ranges, and Graphs of Functions in MTH 261 Final Exam and more Exams Analytical Geometry and Calculus in PDF only on Docsity!

Helpful Concepts for MTH 261 Final

What are the general strategies for determining the domain of a function?
How do we use the graph of a function to determine its range?
How many graphs of basic functions have you already remembered without having to plot points?
How do we recognize an even function? What’s special about an even function in terms of its graph?
How do we recognize an odd function? What’s special about an odd function in terms of its graph?
What is the general procedure in graphing a piecewise-defined function?
How do we recognize functions whose graphs can be obtained from the graph of a known function via vertical/horizontal shifts?
What are the two basic types of graphs for exponential functions?
What are the basic exponential rules?
How do we set up an exponential function if its doubling time (or half-life) is known?
How are the graphs of two functions related if they are inverses?
What are the two basic types of graphs for logarithmic functions?
What are the basic logarithmic rules?
How do we use a logarithm to solve an exponential equation?
What are the definitions of the six trigonometric functions?
How many of the trigonometric values of special angles have you already remembered without having to use a graphing calculator?
What are the three fundamental trigonometric identities?
How are the inverse trigonometric functions defined?
What is the range of each inverse trigonometric function?
Does the existence of the limit lim (^) x → c f ( x )depend on whether the function f ( x ) is defined at

x = c? Does it depend on the value f ( c ) is the function f ( x ) is defined at x = c?

What are the general strategies for computing limits such as lim (^) x → c f ( x )?
If lim (^) x → c f ( x )= L is known and a specific ε > 0 is given, how do we find a suitable δ > 0 so

that the “ f ( ) x − L < ε whenever x − c < δ?

What are one-sided limits? How are they related to two-sided limits?
Do limits generally commute with operations such as addition, subtraction, multiplication, division, and exponentiation?
Can limits of polynomials be found by substitution? Any exceptions?
Can limits of rational functions be found by substitution? Any exceptions?
What does the Sandwich Theorem say? When is it useful?
If lim (^) x →∞ f ( x )or lim (^) x → −∞ f ( x )exists, what does it mean algebraically? What does it mean

geometrically?

If lim (^) x → c + f ( x )=∞(or -∞) OR lim (^) x → c − f ( x )=∞(or -∞) exists, what does it mean

algebraically? What does it mean geometrically?

What are the general rules on ( )

lim Qx

P x x →∞ and^ ( )

lim Qx

Px x → −∞ , where^ P ( x ) and^ Q ( x ) are

polynomials?

What is the precise definition for a function f ( x ) to be continuous at x = c?
Which types of functions are always continuous everywhere?
Which types of functions are continuous wherever they are defined?
Does continuity generally commute with operations such as addition, subtraction, multiplication, division, exponentiation, and composition?
What does the Intermediate Value Theorem say? How is it useful?
How do we determine the tangent line to a point on the graph of a function?
Why does the limit h

f a h f a h

lim (^0)

→ , if exists, give us the slope of the tangent to the point ( a , f ( a )) on the graph of the function f ( x )?

Note that there are three parts in the Chain rule. If you know any two of them, can you solve for the third one?
How do you calculate dy/dx at a given point on a parametrized curve x = x ( t ), y = y ( t )?
How do you calculate d^2 y/dx 2 at a given point on a parametrized curve x = x ( t ), y = y ( t )?
What is implicit differentiation? What are the circumstances when implicit differentiation is needed?
What are the important things to keep in mind when executing implicit differentiation?
What is the normal to a curve at a given point? How do you find its equation?
What are related-rates equations/questions?
Any general strategies to solve related-rates problems?
What are the derivatives of all six inverse trigonometric functions?
How are the derivatives of a pair of inverse functions related?
Let f and g be any two trigonometric functions. How do you compute f ( g -1^ ( x ))?
For any a > 0 and a ≠ 1, what is the derivative of the exponential function ax? What is the derivative of af ( x )^ if f ( x ) is another differentiable function?
For any a > 0 and a ≠ 1, what is the derivative of the logarithmic function loga x? What is the derivative of log a f ( x ) if f ( x ) is another differentiable function?
How do you differentiate functions such as f ( x ) g( x )^ if f ( x ) and f ( x ) are two differentiable functions?
lim x ax → x

0

What is law of exponential change?
What are absolute extrema? Local extrema? Are absolute extrema always local extrema?
What does the Extreme Value Theorem for Continuous Functions say?
What are critical points? Do critical points always give rise to local extrema?
How do you find absolute extrema on a closed interval? What if some critical points are outside of the interval? What if there are no critical points inside the interval?

What does Rolle’s Theorem say?
What does the Mean Value theorem say? What is its geometric interpretation?
List all functions whose derivative is constantly zero.
How are two functions related if they have the same derivative?
If the acceleration of a moving object is a constant a , what do its velocity function and position function look like?
Why is a function increasing on an interval if its first derivative is positive on this interval? Similarly, decreasing when negative?
What does the First Derivative Test for Local Extrema say?
When is a function’s graph concave up/down on an interval? Give a geometric reason.
What is a point of inflection?
What does the Second Derivative Test for Local Extrema say?
Summarily, how do we learn about the graphic shape of a function from its derivatives?
What is your general strategy for solving max-min problems? Give examples.
Describe the linearization of a function at a given point. What’s its geometric interpretation?
If y = f ( x ) is a differentiable function, describe the differential dy (= df ).
How do we use the linearization or differential of a function to estimate its values?
Describe Newton’s Method. What’s its use?
Table 4.1 (p. 335). What is the reason to include an additional “ C ” in each of the formulas?
What is an antiderivative? Indefinite integral?
What is an initial value problem? What’s your strategy to solve an initial value problem?
Describe the concept of integration by substitution. How does it work?
Describe the Definite Integral as a limit of Riemann sums.
How can geometric formulas be used to compute definite integrals? Give a few examples.

What is your general strategy in deciding on which method to use?
How do we compute the arc length of a curve y = f ( x ). Interpret the factor 1 + [ f ' ( )] x^2.
Regarding arc length, what if the curve is given by x = g ( y )?
Regarding arc length, what if the curve is given by parametrization^ x^ =^ α^ ( t ),^ y^ =^ β^ ( t )?
How do we compute the centroid of a region between two curves?
How do we compute the center of mass of a region between two curves?
When would the centroid and the center of mass of a plane region be the same?
How do we compute the total mass of a plane region?
How do we compute the moment of a plane region about the x -axis or y -axis?

x x x

r  

lim (^) → ∞ 1 =? lim 1

kx x

r hx →∞

Strategies for Determining Domains, Ranges, and Graphs of Functions in MTH 261 Final Exam, Exams of Analytical Geometry and Calculus

Related documents

Partial preview of the text

Download Strategies for Determining Domains, Ranges, and Graphs of Functions in MTH 261 Final Exam and more Exams Analytical Geometry and Calculus in PDF only on Docsity!