GENERAL TIPS WHEN TAKING THE AP CALC EXAM. Multiple Choice Portion

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1 GENERAL TIPS WHEN TAKING THE AP CALC EXAM. Multiple Choice Portion 1. You are hunting for apples, aka easy questions. Do not go in numerical order; that is a trap! 2. Do all Level 1s first. Then 2. Then 3. a. Level 1 questions are questions that enable you to have quick auto-responses. i. You immediately know what it s asking you to do and ii. You know that getting the answer takes minimal effort. b. Level 2 questions are questions that you know how to solve, but requires time/effort. i. I know how to do this, but this will take some time. c. Level 3 questions are questions that you don t immediately know how to do. i. Save these questions for the very end. ii. These are the questions that kill a lot of time. Free Response Questions (FRQs) 1. Don t box anything. If you do, AP graders are limited to only grading what you boxed. 2. Equal is equal. No need to simplify. a. For example, if you are asked to differentiate a function that requires the quotient rule. i. Setup the quotient rule, but that s it. You will not get extra points for simplifying. 3. You are still on the hunt for apples. Look for the easiest problems. a. Just because you don t know part a, doesn t mean you can t do part b, c, d. i. This, too, is a trap! 4. During the calculator portion, there s no need to show work. a. For instance, you may be required to setup an integral for a problem. i. Setup integral. Use a calculator. 1. There s no need to show the actual integration. Graphing Calculator 1. Be quick. Practice your common functions now. a. We can: i. Differentiate given a value ii. Integrate given a value iii. Calculate 1. Intersections 2. Zeros 3. Max/Min 2. Always round to three decimal places. You will lose key points for not doing this.

2 What do I need to get at least a passing score of 3? points are available in MC. 54 points in FRQs. a. Each equally weighted 50% each. 2. Approximately 36% will get you an AP 3 score. 3. There are 45 MC questions. Each worth 1.2 points. Total 54 points. 4. Each FRQ is 9 points. 6 FRQs. 54 available points. a. It s OK to not get all 9 points. b. Theoretically, if you can get at least 3 out of 9 points, you re in good shape. 5. Scenarios: a. If you get 0 FRQ points, but get 34 MC questions = AP SCORE 3. b. If you get 0 MC questions, but get 41 FRQ points = AP SCORE 3. c. IF you get 3 pts per FRQ (18 points), and 19 MC questions = AP SCORE So what? a. The more MC Questions right, the more room you have with FRQs. b. If you get 25 MC questions correct (out of 45), you ll just need 11 FRQ points (out of 54). i. Meaning: You can get 20 MC questions wrong and lose 43 FRQ points and still pass with a 3!

3 AP Calculus AB Summary Big Idea #1: Limits o Limits only exist when the left and right limits are equal. o Limits à when x gets arbitrarily close, but not exactly, to a. o Defined functions à when x is exactly at a. o Algebraically Always directly substitute if possible. If not, manipulate until you can substitute. NOTE: If you substitute and get 0/0 (indeterminate) know that you have the option of applying L Hospital s Rule. Differentiate the top and the bottom. (THIS IS NOT the QUOTIENT RULE.) Then plug in a. o If your answer still comes out 0/0, keep repeating until it doesn t. Sometimes indeterminate can look like infinity/infinity. o Special Limits to Remember o Vertical asymptote à as x approaches a, the limit is +/- infinity. When your denominator = 0. o Horizontal asymptote à as x approaches infinity, the limit is a. Divide every term by the largest degree. All remaining fractions will go to zero (because when denominators get large, fractions go to zero). o Continuity When the limit exists and is equal to the defined function. Informally, a function is continuous when you don t lift up your pencil. Generally, any function is continuous as long as a is in your domain. 3 Types of Discontinuity Hole (when you cross out a factor) Infinite (vertical asymptotes) Jump (piecewise functions) o Intermediate Value Theorem (IVT) Destiny s height from birth to now example. If f is continuous in [a,b], then there exists a c in [a,b] such that f(c) is within [f(a), f(b)].

4 Big Idea #2: Derivatives o Average rate of change (TOLL BOOTH) m = (y2-y1)/(x2-x1) aka secant line o Instantaneous rate of change (COP SCENARIO) Also known as Derivative Slope of a curve Tangent o Limit Definition of a Derivative Just know the skeleton of it, and recognize that it s just fancy for derivative.!"#!!!!!"# (!) lim!!! Answer: cos(x). lim!!!!!!!!!! means what s the derivative of sin(x)? means what s the derivative of x^2? Answer: 2x. o (NOTE: You can even apply L Hospital if you forget.) o If D, then C. (If f is differentiable, then it is continuous at a.) o Not Differentiable at Corners! at a discontinuity! at a vertical tangent (because slope is undefined). o Notation!!! means the fifth derivative.!!! o Derivative Formulas Memorize ALL of them. Don t forget trig, trig inverse, exponentials, logs. Product Rule 1 st times derivative of the 2 nd, plus 2 nd times derivative of the 1 st. Quotient Rule (Drake s Rule) Bottom times derivative of the top, minus the top times derivative of the bottom, all over bottom squred. Chain Rule. Derivative of the outside, of the inside, times derivative of the inside. o Implicit Differentiation When you have an equation in terms of both x and y Take the differential (with respect to that variable). o Then divide all by dx. o Isolate dy/dx. o Derivative of an Inverse Function. Label out f and g. Jot down given point. o Check notes for remaining steps.

5 o Find the equation of the tangent line You need a slope and point. Point is usually given. Differentiate, to find general slope, then plug in given point. Plug your point and slope into point-slope form. Parallel à Same Slope Normal à Perpendicular Slopes (opposite reciprocal slopes). o Related Rates Always give you the rate of one quantity that s changing Asks you to find the rate of something else that s changing. General steps Given rate Asks for another rate. Find an equation that relates the two rates. Differentiate, Plug, Solve, Provide Units. NOTE: If it involves a triangle consider: Pythagorean Theorem Similar Triangles (Proportions) SOH-CAH-TOA. o Linear Approximations Equation of a tangent line (but in slope-intercept form) Y = y1 + m(x-x1) Helps approximate the value on a curve. o Extreme Value Theorem (Absolute) If f is continuous on [a,b], then f attains an absolute max and min somewhere on or inside the bounds of [a,b]. Get critical numbers. Plug in critical numbers AND bounds into function. Your highest value would be considered your abs max. Your least value would be considered your abs. min. o Critical Numbers Take the derivative, and find your zeros AND undefined values à these are your critical numbers. o Sign Chart Number line that consists of your critical numbers. Make sure YOU label your sign chart. Does it represent f or f? Make sure YOU label what a + or a represents (i.e. f is decreasing.) THEY ARE PICKY. BE DETAILED.

6 o f vs. f vs. f With f, you can get the following information regarding f. When f is +, original f is increasing. When f is -, original f is decreasing. When f changes from + to -, there exists a local max. When f changes from to +, there exists a local min. With f, you can get the following information regarding f. When f is +, original f is concave up. When f is -, original f is concave down. When f changes signs (aka concavity), there is a point of inflection. If you re looking at a given f graph, POI is where the concavity literally changes. If you re looking at a given f graph, POI is the local extrema. (read below) If you re looking at a given f graph, POI is the x-intercept (when sign, aka concavity, changes). With f, you can get the following information regarding f. When f is +, f is increasing. When f is -, f is decreasing. So if you re looking at a given f graph, its local extremas are actually points of inflection (because extremas are moments when something starts and stops increasing/decreasing. o Mean Value Theorem If f is continuous on [a,b] and differentiable on (a,b), then there is a number c in (a,b) such that f! c =!!!!(!)!!! Aka (secant line parallel to the tangent line) o Rolle s Theorem If f is continuous on [a,b] and differentiable on (a,b), AND f(a) = f(b), then there is a number c in (a,b) such that f! c = 0 Aka (secant line comes out 0, so the slope of the tangent line must = 0 as well) AKA Special Case of MVT. o Optimization When you want to maximize/minimize something. Usually involves two equations. One equation is to help you substitute for another variable. 2 nd equation is to differentiate. o If you want to optimize area, differentiate the area equation.

7 Big Idea #3: Integrals o AKA antiderivative o Area Approximation Left Riemann Sum Use the left-most points Start with first x-coordinate, end with second to last. Right Riemann Sum Use the right-most points Start with second x-coord, end with last. Midpoint Rule Use the averages between two x-coords. Trapezoidal Rule In-between bases get doubled. Only works when equally partitioned. Over/Underestimates Draw it out to see if the estimated area is under/over the curve. Area BELOW the x-axis is NEGATIVE area. o Integral Exact Area (because number of rectangles go to infinity) o Fundamental Theorem Of Calculus FTC1 When you take the derivative of an integral. Conditions that must be met: o Lower bound must be a constant. o Upper bound must be a function. Once conditions are met, o FTC1 tells you to plug in upper bound into function. If the upper bound is any other function than x, you must apply the chain rule. Check notes. Integral Properties to consider if conditions are NOT met:

8 Know the meaning behind the derivative of an integral: FTC2: When you re taking an integral that is bounded: For unbounded integrals, don t forget + C!!!!!! Memorize all Integral Formulas. ALL common trig integrals Exponential/Logs Trig Inverses Position vs. velocity vs. acceleration IMPORTANT: This concept can be applied to both Derivatives and Integration. o It depends on what s given and what s needed. I.e. if they provide position and ask for velocity. Derivative. If they velocity and ask for position Integration. Speed: Is velocity. Same as velocity, but always positive. Distance vs. Displacement o Distance total traveled. (always positive) o Displacement Where you started to where you ended. U-substitution Look for two expressions that are one degree from each other. U is usually the function inside of another function. If you re dealing with trig functions, you must strategically decide what s u and what s du. Remember that bounds will change if you use u-substitution.

9 o Area between curves. Top minus bottom (with respect to x). Right minus left (with respect to y). If equations are in terms of y, you will most likely integrate with y. o Volume For Disk/Washer/Shell If possible, always try to get a vertical rectangle so you can integrate with respect to x. Depending on the axis of rotation, o if it is perpendicular to your vertical rectangle, use DISK/WASHER. o If it is parallel to your vertical rectangle, use SHELL. o PerpenDISKular vs. ParaSHELL. Know how to modify your lengths when the axis of rotation is a line that s NOT the x and y-axis. (i.e. about the line x = 20 or y = -5) Disk/Washer Disk if it s completely hugging the axis of rotation. Washer if it is not (creating a hole). Has a pi coefficient on the outside of the integral. Shell Has 2pi coefficient on the outside of the integral. Find (shell radius)(shell height) Cross Sections. There is NO axis of rotations. They are just stacking the same shape, where the size depends on the height of the theoretical rectangle. Perp to the x-axis à Integrate with x. Perp to the y-axis à Integrate with y. Remember all of your Area formulas. Make sure you understand the difference between semicircle and quarter circle. o What does the theoretical rectangle aka height represent, in regards to both shapes? Diameter. Radius. Respectively. o Average value of a function The value that gives you the same amount of area (in a rectangle) as it is under its curve. 1/(b-a) outside the integral. Inside the integral: height.

10 o Slope Fields A field of slopes at particular points. Plug into differential. Sketch the slope. All slopes have to relatively steeper/less steep from each other. o Exaggerate. When you re matching slope fields with differential equations. Look for zero slopes and/or undefined slopes. Pick a point that is unique from the rest to help with elimination. When you re matching slope fields with original equations. Look at the general flow of the slopes to see what function it shapes like. o Separable Equations Product/Quotient? If not, manipulate until it does look like one. Separate. Integrate, don t forget +C. Know how to handle natural logs, exponentials, laws of exponents/logs. They will judge you on how well you handle the +C.