Review Guideline for Final

Transcription

1 Review Guideline for Final Here is the outline of the required skills for the final exam. Please read it carefully and find some corresponding homework problems in the corresponding sections to practice. Section 1.1: Limits: An Intuitive Approach Given the graph of a function f(x), be able to understand the graph and answer questions like: 1. Find the function value at a point 2. Find the left side limit and the right side limit 3. Determine whether a limit exists 4. Find the limits at infinity 5. Indicate whether the function has any horizontal asymptotes or vertical asymptotes Section 1.2: Computing Limits 1. For a limit of type or, be able to simplify/rewrite the original function to find the limit. For example, using complete squares or 2. For a piece wise defined function, be able to answer the following questions: a. Find the function value at a point b. Find the left side limit and the right side limit c. Determine whether the limit exists by using the left side and right side limits d. Continuity of a piecewise defined function (combined with Section 1.5: Continuity) Section 1.3: Limits at Infinity 1. Be able to find the limits of rational functions at infinity. 2. Be able to find the limits of functions involving radicals.

2 Section 1.5: Continuity 1. Know what it means for a function to be continuous at a specific point and on an interval. 2. Know the continuity of polynomial functions, rational functions, exponential functions, logarithmic functions and composited functions. 3. Use the continuity property of a given function to calculate a limit. 4. Determine the continuity/discontinuity of a piecewise defined function (combined with Section 1.2: Computing Limits). Section 1.6: Limits & Continuity of Trigonometric Functions 1. Know where the trigonometric and inverse trigonometric functions are continuous and be able to use the continuity of those functions to calculate limits. 2. Be able to use lim lim 1, lim 1 and 0 to help find the limits of functions involving trigonometric expressions, when appropriate. Section 2.1: Tangent Lines & Rates of Change (the questions might be combined with Section 2.2) 1. Be able to compute the average rate of change of a function over an interval, or equivalently, be able to find the slope of the secant line through two points on the graph of a function. 2. Be comfortable using a limit to compute the instantaneous rate of change of a function, or equivalently, know how to find the slope of the tangent line to a function Section 2.2: The Definition of the Derivative 1. Know how to compute the derivative of a function using the limit definition. 2. Be able to use the derivative to help find an equation of a tangent line. 3. Be able to match graphs of functions with the graphs of their derivatives.

3 Section 2.3&2.4: Techniques of Differentiation 1. Know that the derivative of a constant function is Be able to compute the derivative of a power function. 3. Be able to compute the derivatives of constant multiples, sums, differences, products and quotients of differentiable functions. 4. Be able to compute higher order derivatives Section 2.5: Derivatives of Trigonometric Functions 1. Be able to know the derivative of the 6 elementary trigonometric functions. 2. Be able to apply the product rule, quotient rule and chain rule (in Section 2.6) to find derivatives of functions involving trigonometric functions. Section 2.6: The Chain Rule Be able to know how to use the chain rule to calculate derivatives of compositions of functions, such as, power functions, trigonometric functions, exponential functions and logarithmic functions. Section 3.1: Implicit Differentiation 1. If two quantities and are related in an equation, be able to differentiate the equation to obtain a new equation in terms of, and. 2. Be able to solve this new equation for which can be represented in terms of and. 3. Be able to further find. Section 3.2: Derivatives of Logarithmic Functions 1. Be able to compute the derivative of the logarithmic function ln. 2. Be able to compute the derivative of the compositions of functions involving logarithmic functions. 3. Know how to use logarithmic differentiation to find the derivative of functions involving products and quotients.

4 Section 3.3: Derivatives of Exponential and Inverse Trigonometric Functions 1. Be able to compute the derivative of the exponential function. 2. Be able to compute the derivative of the compositions of functions involving exponential functions. 3. Be able to compute the derivatives of the inverse trigonometric functions, specially, sin, cos, tan and cot. 4. Be able to compute the derivative of the compositions of functions involving inverse trigonometric functions. 5. Know how to apply logarithmic differentiation to compute the derivatives of functions of the form, where and are non constant functions of. Section 3.4: Related Rates Be able to solve related rates problems. Read the strategy provided in the list of exercises of Section 3.4. Section 3.5: Local Linear Approximation 1. Be able to find the local linear approximation of a given function at a specific value. 2. Know how to use the local linear approximation to estimate a given quantity. Section 3.6: L Hoptial s Rule 1. Know how to use L Hopital s Rule to compute limits involving indeterminate forms of types and. 2. Be able to compute limits involving indeterminate forms of types, 0, 0, and 1.

5 Sections 4.1 and 4.2: Increasing, Decreasing & Concavity 1. Know how to use the first derivative to find the critical points of a function 2. Know how to use the signs of the first derivative of a function to find intervals on which the function is increasing or decreasing. 3. Know how to use the second derivative of a function to find intervals on which the function is concave up or concave down. 4. Know how to find the locations of inflection points. 5. For a critical point, be able to apply the First Derivative Test and Second Derivative Test (when appropriate) to determine if the critical point is a relative maximum, relative minimum, or neither. Sections 4.2 and 4.3: Sketch the Graphs of Functions For a given function, 1. Be able to find its domain, x intercept(s), y intercept; 2. Be able to label all horizontal asymptotes and vertical asymptotes if any; 3. Be able to label the coordinates of all critical points if any, and be able to find the intervals on which the function is increasing and decreasing; 4. Be able to find all the relative extrema; 5. Be able to find the intervals on which the function is concave up and concave down, and be able to label the coordinates of all inflection points; 6. Be able to use all the above information to sketch the graph of the function. Section 4.4: Absolute Extrema 1. Be able to find the absolute maxima and minima of a continuous function on a finite closed interval. 2. Be able to find the absolute maxima and minima of a continuous function on an infinite interval. Section 4.5: Applied Max/Min Problems (Optimization) Know how to use the techniques from Section 4.4 to solve optimization problems, i.e. given a system of related quantities, find values of the quantities that optimize one of them (e.g. minimize a cost, maximize a volume, etc).