AP CALCULUS AB Study Guide for Midterm Exam 2017 CHAPTER 1: PRECALCULUS REVIEW 1.1 Real Numbers, Functions and Graphs - Write absolute value as a piece-wise function - Write and interpret open and closed intervals - Find the distance between two point - Find the root(s) of an equation - Use the equation of a circle - Apply the vertical line test to see if a graph is a function - Know the difference between even and odd functions - Apply transformations to equations to get other equations 1.2 Linear and Quadratic Functions - State a linear equation in slope-intercept form - Write an equation is standard form, slope-intercept form point-slope form - Find parallel and perpendicular equations of a given equation - Find the roots of a quadratic equation either by factoring or the quadratic formula - Apply the process of completing the square, when needed 1.3 The Basic Classes of Functions - Find the degree and leading coefficient of a polynomial function - Know the domain and range and graphs of: rational function, algebraic function, exponential function, and composite functions 1.4 Trigonometric Functions - Convert an angle from radians to degrees, and vice versa - Find the length of an arc on a circle of radius r - Work with radians more than degrees - Know the 6 basic trigonometry functions and their definitions - Know the basic properties of sine and cosine: periodicity, parity, and basic identities - Know the sine and cosine of the basic functions: 0, /2, /3, /4, and their multiples in the four quadrants 1.5 Inverse Functions - Know if a function is one-to-one or not - Find the inverse of f(x) if it exists - Recognize that f(x) is one-to-one if and only if every horizontal line intersects the graph of f(x) in at most one point - The graph of a function s inverse reflects the graph of f(x) through the line y = x. - Find the domain and the range of the main inverse functions
1.6 Exponential and Logarithmic Functions - Identify and know the domain of the exponential and logarithmic functions - Apply the logarithm laws (4) - Know the domain and range and identify the sinh x and cosh x (hyperbolic functions) and their inverses 1.7 Technology: Calculators and Computers - Set the viewing window correctly for the equation that needs to be graphed - Work in radians and not degrees for trig functions - Find how many roots and where they are - Describe the set of all solutions; know when there are no solutions - Identify asymptotes if working with a rational function CHAPTER 2: LIMITS 2.1 Limits, Rates of Change and Tangent Lines - Identify that the average rate of change is the same as the slope of the secant line - Instantaneous rate of change is the limit of the average rates of change, which is the same thing as the slope of the tangent line - find the velocity of an object in linear motion (same as the rate of change of position) - See that the average rate of change over every interval and the instantaneous rate of change at every point are equal to the slope m 2.2 Limits: A Numerical and Graphical Approach - Find the value of the limit using a table or a graph - Determine the value of a limit using the limit definition - Determine the one-sided limits of specific functions numerically and graphically - Apply the concept that the limit may exist even if f(c) is not defined - Identify the asymptote as the vertical line in the case of a one-sided or two-sided infinite limit 2.3 Basic Limit Laws - Evaluate the limit using the basic limit laws 2.4 Limits and Continuity - Identify a discontinuity, and know the three common discontinuities ( removable, jump and infinite) - Apply the definition of one-sided continuity - Define and apply the concept of continuity - Apply the laws of continuity: polynomials, rational functions, nth-root and algebraic functions, trig functions and their inverses, exponential and logarithmic functions
- Substitution method: If f(x) is known to be continuous at x = c, then the value of the limit of f(x) is f(c) 2.5 Evaluating Limits Algebraically - Use substitution when function is continuous - Recognize when a limit is in indeterminate form, and see if it can be transformed into a new expression that you can find the limit of 2.6 Trigonometric Limits - Recognize when the Squeeze theorem need to be applied 2.7 Limits at Infinity - Calculate the limits at infinity, especially involving rational functions and specific limits - Find the horizontal asymptotes of a rational function 2.8 Intermediate Value Theorem - Show that an equation has at least one solution, according to the Intermediate Value Theorem - Show the existence of zeros CHAPTER 3 DIFFERENTIATION 3.1 Definition of the Derivative - Calculate f (a) using the limit definition - Recognize when the definition of derivative is presented and when a shortcut can be used - Apply the equation of the tangent line in point-slope form 3.2 The Derivative as a Function - Write and recognize different notations for derivative - Use the power rule for derivatives - Compute the derivative and find an equation of the tangent line to the graph at a specific point - Find the derivatives of exponential and logarithmic functions - Remember that differentiability implies continuity, not vice versa 3.3 Product and Quotient Rules - Distinguish and apply the product rule and the quotient rule to problems
3.4 Rates of Change - Connect the instantaneous rate of change as the derivative - Solve linear motion problems, to find distance, velocity and acceleration - Solve marginal cost and gravity problems 3.5 Higher Derivatives - Calculate higher derivatives 3.6 Trigonometric Functions - Calculate the derivatives of the six trigonometric functions 3.7 The Chain Rule - Identify when the chain rule needs to be used - Find the derivative using the appropriate rule or combination of rules - Use the chain rule to find derivatives 3.8 Derivatives of Inverse Functions - Find the derivative of the inverse of functions - Compute the derivative at a point dedicated without a calculator 3.9 Derivatives of General Exponential and Logarithmic Functions - Apply and solve derivative formulas for e x, ln x, b x and log xb - Find the derivatives of hyperbolic functions and inverse hyperbolic functions 3.10 Implicit Differentiation - Use implicit differentiation to compute the derivative when x and y are related by an equation - Remember to include the factor dy/dx when differentiating expressions involving y with respect to x 3.11 Related Rates - Calculate an unknown rate of change in terms of other rates of change that are known, including the sliding ladder problem, or filling a rectangular tank or conical tank, tracking a rocket, finding the change in volume of a sphere, the Pythagorean Theorem, similar triangles, etc. - Draw a diagram if possible - Find an equation that relates the known and unknown derivatives
CHAPTER 4 APPLICATIONS OF THE DERIVATIVE 4.1 Linear Approximation and Applications - Use linear approximation to estimate a small change 4.2 Extreme Values - Find the minimum and/or the maximum value of a function - Find the local extrema and critical points - Find the extrema on a closed interval (include the endpoints when calculating) - Recognize Rolle s Theorem and when it can be used 4.3 The Mean Value Theorem - Verify the Mean Value Theorem with a function and endpointe - Find intervals where a function is increasing and decreasing - Analyze and test critical points - Apply and analyze the First Derivative Test 4.4 The Shape of a Graph - Test for Concavity at the points of inflection - Find the points of inflections and intervals of concavity - Know what to do where the second derivative does not exist - Apply and analyze the Second Derivative Test - Find the intervals on which the function is concave up or down, the points of inflection, the critical points and the local minima and maxima - Sketch the graph of a function knowing the critical and inflection points 4.5 L Hopital s Rule - Know when to apply the rule - Know when the rule needs to be applied more than once - Evaluate limits using L Hopital s Rule 4.6 Graph Sketching and Asymptotes - Focus on transition points to sketch a graph by determining the signs of f and f, noting transition points and sign combinations, drawing arcs of appropriate shape and asymptotic behavior 4.7 Applied Optimization - Choose variables and write the objective function and then apply techniques to solve: find the critical points, evaluate the function at the critical points and the endpoints; state the largest and smallest values as the extreme values of f(x) on [a,b]