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 eponential functions? What are the basic eponential rules? How do we set up an eponential 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 eponential equation? What are the definitions of the si 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 eistence of the limit lim c f ( ) depend on whether the function f() is defined at = c? Does it depend on the value f(c) is the function f() is defined at = c?
What are the general strategies for computing limits such as lim f ( ) c? If lim c f ( ) = L is known and a specific ε > 0 is given, how do we find a suitable δ > 0 so that the f( ) L < ε whenever 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 eponentiation? Can limits of polynomials be found by substitution? Any eceptions? Can limits of rational functions be found by substitution? Any eceptions? What does the Sandwich Theorem say? When is it useful? If lim f ( ) or lim f ( ) eists, what does it mean algebraically? What does it mean geometrically? If lim + f ( ) = (or - ) OR lim f ( ) = (or - ) eists, what does it mean c c algebraically? What does it mean geometrically? What are the general rules on polynomials? lim P( ) and Q( ) lim P( ), where P() and Q() are Q( ) What is the precise definition for a function f() to be continuous at = 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, eponentiation, 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? f ( a + h) f ( a) Why does the limit lim h 0, if eists, give us the slope of the tangent to the h point (a, f(a)) on the graph of the function f()?
f ( a + h) f ( a) In computing lim h 0, if it eists, what do you generally epect to happen to h the original quantity h in the denominator? What is a parametrized curve in the y-plane? What are the general strategies to find a Cartesian equation for it? What is the standard parametrization for the circle 2 + y 2 = a 2? For ( 2 / a 2 ) + (y 2 / b 2 ) = 1? What is the instantaneous rate of change of a function f() at = c? How is it related to the derivative of f() at = c? How are position, velocity, and acceleration related? What are marginal revenue, marginal cost, and marginal profit functions? What does the value of a marginal function at a certain production level predict? How do you differentiate an arbitrary power of the variable? How do you differentiate a product of several functions? How do you differentiate a quotient of two functions? How do you differentiate a power of a function? What are the derivatives of all si trigonometric functions if the variable is measured in radians? What are the derivatives of all si trigonometric functions if the variable is measured in degrees? 2006 d cos =? 2006 d lim sin 2 sin 3 0 =? Is there a value of c that will make f ( ) = if 0 2 continuous at c if = 0 = 0? lim cos 1 0 =? Is there a value of b that will make + b if < 0 f ( ) = cos if 0 differentiable at = 0? f ( a+ h) f ( a h) If a function f() is differentiable at = a, then limh 0 =? h How do you differentiate the composite of two functions? Three functions?
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/d at a given point on a parametrized curve = (t), y = y(t)? How do you calculate d 2 y/d 2 at a given point on a parametrized curve = (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 eecuting 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 si 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 ())? For any a > 0 and a 1, what is the derivative of the eponential function a? What is the derivative of a f() if f() is another differentiable function? For any a > 0 and a 1, what is the derivative of the logarithmic function log a? What is the derivative of log a f() if f() is another differentiable function? How do you differentiate functions such as f() g() if f() and f() are two differentiable functions? a 1 lim 0 =? What is law of eponential change? What are absolute etrema? Local etrema? Are absolute etrema always local etrema? What does the Etreme Value Theorem for Continuous Functions say? What are critical points? Do critical points always give rise to local etrema? How do you find absolute etrema 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 Etrema 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 Etrema say? Summarily, how do we learn about the graphic shape of a function from its derivatives? What is your general strategy for solving ma-min problems? Give eamples. Describe the linearization of a function at a given point. What s its geometric interpretation? If y = f() 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 eamples.
How can definite integrals be used to compute summations? Give a few eamples. How do we determine the average value of an integrable function on an interval? Give an intuitive interpretation of the formula. b a If f() 0 on [a, b] and f ( ) d= 0, what can you can about the function f() on [a, b]? What does the Mean Value Theorem for Definite Integrals say? What is the geometric interpretation of this theorem? If f() is a continuous function on an interval [a, b] and a f ( ) 0, must f() have a zero in [a, b]? Why or why not? What does the Fundamental Theorem of Calculus say? Give both versions. What does the Fundamental Theorem of Calculus say about the relationship between differentiation and integration? If f() is a continuous function, and both α() and β() are differentiable functions, then d α ( ) β ( ) f () t dt=? d d What is the geometric interpretation of the general fact that a f () t dt = f( )? d When changing the variable of a definite integral, how are the upper and lower limits of the integral changed? What s the advantage of changing these limits? How do we compute the area of the region between curves? Must we always integrate with respect to? Is there a scenario that we may have to set up more than one integral to accomplish the task? What is numerical integration? Describe the Trapezoidal Rule. Describe Simpson s Rule. Is there a special condition on the number n of partition points? Interpret the fact that integrating the cross-section area function of a solid provides the volume of this solid. Interpret the Disk Method. Does your description depend on the ais of revolution? Interpret the Washer Method. How is it related to the Disk Method? Interpret the Shell Method. Does your description depend on the ais of revolution? b
What is your general strategy in deciding on which method to use? 2 How do we compute the arc length of a curve y = f(). Interpret the factor 1+[ f '( )]. Regarding arc length, what if the curve is given by = g(y)? Regarding arc length, what if the curve is given by parametrization = 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 -ais or y-ais? r r lim 1 + =? lim 1± h k =?