AP Calc AB Notes PDF

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Notes for AP Calculus AB using the Calculus of a Single Variable 10e textbook. These notes outline the process to do things in AP Calculus AB....

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AP Calculus Notes: AB: Pre-calc Stuff: ● Know all of this (it’s vital): (insert quizlet link) Limits: ● If the limit of a function as x approaches a constant (c) exists then the function is interrupted. ● Types of Discontinuities: ○ Hole (removable) ○ Jump Discontinuity (non-removable) ○ Infinite Discontinuity (non-removable) ● If limit from the right isn’t the same as the limit from the left then the limit does not exist (DNE). ● Limit of a constant is the constant itself. ● Plug in the value for x (c) to solve for basic limits (duh). ● To solve limit questions where you get the indeterminate form of 0/0 or ∞/∞ you can: ○ SUBSTITUTE (ALWAYS CONSIDER THIS FIRST! But since you got the indeterminate form move on to the next steps) ○ Factor ○ Rationalize ○ Simplify/Use Identities ○ Common denominator (usually for numerator when there are fractions).

● Keep simplifying until something can be plugged in. ● When trying to simplify a limit sometimes your can multiply individual terms that are multiplied together by “1” in a sneaky way. You can multiply one term’s denominator by “x” and the other term’s numerator by “x”. x/x=1, so this is legal. ● See “Properties of limits” section on identities sheet. Take the limit of the function and then do whatever else (square root, raise to power, add, subtract, multiply, divide, etc. ● ●

(This is very important!) ○ Remember this: ■ https://photos.app.goo.gl/8rN3dVxkSdMX3Z St9

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that a^4-b^4 . ● Check out ● When given the limit as h approaches 0 with the definition of the derivative as the function then just take the derivative of the stuff after the “-” in the numerator (That’s the function). ● Use BOB0, BOTNA, and EATSDC to find horizontal

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asymptotes. When finding horizontal asymptotes check the limit from both sides (+ and -) when the infinities are close (tied). If Bigger On Top by 1 then divide the functions to find the oblique asymptote. Don’t forget your identities when solving limit problems!!! They are incredibly useful! When using the definition of the derivative plug in (x+h) in for every x in the function. For the alternative form, use C in the bottom and F(C) in the top!!! Only cancel things out when they are a product of something/not being subtracted. Always distribute the negative!!! Always check for this! Check your factoring! Check the limit from both sides! Especially when given a piecewise function!

Continuity: ● Three conditions must exist to determine whether the graph of the function is continuous at x as it approaches c: ○ The function is defined at x = c. ○ The limit of the function exists at x = c from both sides. ○ The limit of the function exists at x = c and equals

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f(c). Reminder of Types of Discontinuities: ○ Hole (removable) ○ Jump Discontinuity (non-removable) ○ Infinite Discontinuity (non-removable) ■ For example the volcano graph, y=1/x^2 If the limit does not exist (from both sides) there’s a jump discontinuity and your work is done. If the limit exists, continue to see if it equals the function at that x-value (c). If it does the function is continuous but if it doesn’t then the function has a jump discontinuity. When dealing with absolute value functions plug in xvalues to figure out the shape of the graph and then you can find the limit. When checking functions with a radicand, the radicand should always be positive or equal to 0. ○ Solve the radicand function ○ Should get an answer with + or -. ○ Check number within each region to find which areas are defined. ○ Establish the domain and then proceed to answer the question. When checking continuity on a closed interval: ○ Check the continuity from the right of the x-value (in the ordered pair). ○ And the the left of the y-value (in the ordered

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pair) ○ If both limits are the same then the function is continuous on the interval. If “b” is a real number and “f” and “g” are continuous at x = c, then... ○ Bf (number times a function) ○ Fg (multiply functions) ○ F+or-g (adding or subtracting functions) ○ f/g (g can’t equal 0) (dividing functions) ○ F of g(x) (composite functions) ○ are also continuous. The Intermediate Value Theorem (IVT): ○ A mysterious y-value that appears between xvalues. ○ The IVT is sometimes used to find a function’s zeros. ○ To find/use the IVT…: ■ Factor the function and set it to 0. ■ Find the roots and only choose the ones that zare in the interval specified. When finding Vertical Asymptotes (V.A.s) set the denominator to 0. ○ If you canceled out a term in the numerator and the denominator then that can’t be a V.A. Extreme Value Theorem: ○ There are always maximums and minimums on a continuous function on a closed interval.

● Infinite Limits: ○ The limit of 1/x as x approaches 0 from the right, increases without bound (+∞). ○ The limit of 1/x as x approaches 0 from the left , decreases without bound (-∞). ○ When the limit of a function as x approaches a from the left or the right equals positive or negative infinity there’s a V.A. ■ x=a is a vertical asymptote of y=f(x). ○ To do these types of problems: ■ Set denominator to 0. ■ Take the limit of those values. ■ If you get a positive number/0 that’s undefined and going towards positive infinity. ■ If you get a negative number/0 that's undefined and going towards negative infinity. ■ If the limits don’t match then the limit does not exist (DNE) and there’s possible a V.A. and there’s a discontinuity (if you canceled out terms then there’s a hole). ○ 1/+or-∞ = 0 Derivatives: ● Concept (2-1): ○ Secant line touches at two points (average rate of

change). ○ Tangent line touches at one point (instantaneous rate of change). ○ Definition of the derivative (slope of tangent line): ■ ■ When using the definition of the derivative plug in (x+h) in for every x in the function. ■ Don’t forget to distribute the negative! ○ The alternative form of the derivative (finding the average rate of change):

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■ ■ For the alternative form, use A in the bottom and F(A) in the top!!! Once you simplify the equations for the definition of the derivative or the alternate form you must plug in an x-value (c) where you want to find the instantaneous slope. When asked for the equation of the tangent line use point slope form (y-y1)=m(x-x1) to easily write the equation. ■ You may have to change this to standard form by subtracting y1 and then distributing m. f’(x) is first derivative f”(x) is second derivative or the derivative of the

derivative. ■ This can also be written as: ● d^2y/dx^2 ● F^(2)(x) ○ Differentiability = Whether Derivative Exists ○ Sharp turns or cusps cause problems for the derivative. ■ Justification for sharp turn can be like this: ● ○ Continuity does not guarantee differentiability (but it can imply it, don't rely on this!). ○ Differentiability (as long as it’s not a piecewise function) guarantees differentiability. ■ If you run into a piecewise function confirm continuity then prove differentiability. ■ When checking differentiability for piecewise functions the derivatives form both sides must match for it to exist and the function to be differentiable at that point. ○ Graphs of the Derivatives of Functions: ■ Look at graphs and assign tangent lines. ■ Look at what those tangent lines are doing and label positive or negative slope. ■ Label if slope is zero ■ Concentrate on what slope is at 0. ■ These slopes will translate to values on the graph of the derivative.

● Positive slope=positive values on derivative graph. ● Negative slope=negative values on derivative graph. ● Zero slope=crosses x-axis on derivative graph. ● If slope is positive at 0 then the graph of the derivative should be above the xaxis and visa versa. ● Derivative Rules (2-2): ○ Linearization (Not in this section but on getafive): ■ Linearization=linear approximation ■ If you can come up with a tangent line equation, you can use it to approximate the values of f(x)≈L(x). ● If you’re not given a derivative and/or a point use the equation/x-values given to find it. ● Once you have the derivative and a point, write an equation using pointslope form and change it to standard form. ● Replace y with L(x) and plug in what they want you to find (ex: f(6.2) ) for “x”. ● Solve ○ Derivative of a constant is 0 (2-2). ■

○ The power rule is the fastest way to find common derivatives (2-2): ■ ○ The constant multiple rule (2-2): ■ ■ Derivative of constant times function = constant times derivative of function. ○ Sum and difference rule allows you to take individual derivatives of functions added or subtracted together (2-2). ■ ○ THE PRODUCT RULE: ■ ■ Use whenever you’re trying to take the derivative of two functions being multiplied together. ■ Or you see two variables in a derivative problem. PUT PARENTHESIS AROUND THE VARIABLES BEFORE YOU EVEN START T O INDICATE YOU’RE USING THE PRODUCT RULE. ○ THE QUOTIENT RULE: ■ ● Use whenever you see two functions being divided in a derivative problem.

○ Finding tangent lines: ■ Find the derivative of the equation given at the x-value given. This is your slope ■ Use the x-value given to find the respective y-value by plugging it into the original function. This will give you a point if you weren’t originally given one. ■ Use point slope form (y-y1)=m(x-x1) to make an equation. ■ Change it to slope intercept form if necessary. ● If asked for the normal line just take the opposite reciprocal of the slope you found. ○ Finding the tangent lines of an ellipse: ■ There are four. ● y^2+x^2=1 is the equation for a circle with radius 1(Just an FYI). ● When trying to find multiple horizontal tangent lines: ○ Set derivative of function to 0 ○ Plug what you get into ORIGINAL function to get a point. ○ Slope is 0 because the tangent lines are “horizontal” ○ Use point-slope form to write the equations.

○ Derivative of position, s(t), is velocity, v(t), and derivative of velocity is acceleration, a(t). ■ Direction change occurs when graph crosses x-axis (ex: going from positive to negative) and v(t)=0 ■ Thing is slowing down when slope is negative (mostly) but say when v(t) and a(t) have different signs. Or look at graph and find where this is true. ■ Thing is speeding up when slope is positive (mostly) but say when v(t) and a(t) have the same sign. Or look at graph and find where this is true. ■ Thing is moving right when v(t) >0 ■ Thing is moving left when v(t) 0 ■ Function is decreasing if f’(x) < 0 ■ First Derivative Test: ● If the critical value at f’(x) = 0 has a change from negative to positive it’s (relative)a minimum. ● If the critical value at f’(x) = 0 has a change from positive to negative it’s (relative) a maximum. ■ Turning points on first derivative grade can be points of inflection. ○ Information given by the second derivative (only if on interval): ■ There can be an inflection when signs change, f’’(x) = 0, or f’’ is DNE.

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● Find critical values of the second derivative and make a sign chart with them to find where signs change and where inflections will occur. If f’’(x) > 0 on an interval then the graph is concave upward (tangent line lies below the graph) and there will be a min at (c,f(c)). If f’’(x) < 0 on an interval then the graph is concave downward (tangent line lies above the graph) and there will be a max at (c,f(c)). Concavity can also be found with a graph on the first derivative as someone can find the slope of f’: ● If the slope of f’ is positive the graph is concave upward. ● If the slope of f’ is negative the graph is concave downward. Second Derivative Test for Local Extrema: ● Let f’(c) = 0 ● If f’’(c) > 0 then f has a local max at c. ● If f’’(c) < 0 then f has a local min at c. ● If f’’(c) = 0 the test is inclusive and now you have to do the first derivative test to find extrema. ● JUST USE FIRST DERIVATIVE TEST TO FIND EXTREMA (unless specifically told to second derivative test)

● USE SECOND DERIVATIVE WHEN ASKED FOR CONCAVITY!!! ● May have to find POI in order to begin a problem. ■ Graphing Calculator Tips: ● Normal view window is zoom 6 ● Normal view window for trig problems is zoom 7 ● If interval is given in the problem use that as your window ● When given two functions and asked when they have the same slope find both derivatives and graph them both. Then find the intersection point to arrive at your answer. ○ Using an intersection point can also sometimes be used when there are trig functions in the MVT and you must find “C” (and it’s marked as a calculator problem). ● When it’s a calculator problem about finding the x-value that makes the slope/derivative equal a constant you should: ○ Find the derivative ○ Set it equal to the constant and subtract the constant to the other

side. ○ Now take that function and graph it ○ NOW THE ZERO is the x-value!!! Limits At Infinity: ● When you have a limit at infinity: a) Use BOB0 and EATSDC b) If BOTNA comes up try and use L’Hopital’s Rule: i) Only use L’Hopital’s Rule if you get an indeterminate form and there is NO RADICAL SIGN! ii) Do L’Hopital’s Rule by taking the derivative of the numerator and denominator and seeing if you can plug in the given value (in this case infinity). iii) Repeat until you get an answer. c) You can also use fuzzy math: i) If you have a number over infinity the answer is approximately 0. ii) If you have a infinity over a number the answer is approximately infinity. d) If you come into contact with a limit question with a radical sign (they say) to: i) Divide by something such as √X2 for x in order to simplify.

Optimization Problems: ● Find the shortest distance from the given point to a point on the graph. ○ Use the distance formula and plug in the two known values ○ Try and knock out one variable by making something like this: ■ y=x2 ○ Simplify and raise everything to the one-half power ○ Differentiate ○ Now set only the numerator to 0 ○ Now plug in the x-value you get into the original function to find the corresponding y-value. ○ Now you have your point and you’re done. ● Optimization Real World Problems (shapes): ○ Use the formula for the particular shape or you may have to make up a formula based on the problem ○ Knock out a variable (probably will have to use another formula) ○ Differentiate ○ Set that to 0 and solve for one variable ○ Use that to find the other variables Linearization/Tangent Line Approximation: ● Use the point-slope formula

● You should be given a point and a slope or be able to find it ● Often you will have to know that a point on the inverse of a function is opposite on the original function ○ Ex: (1,2) ---> (2,1) on inverse ● Make the equation ● Plug in the value you are approximating in for X ○ y-1=4(X-3) ● Solve for Y ● You will often be asked if your answer is an overestimate or an underestimate. ● You can figure this out by finding the second derivative and seeing if: ○ F’’ < 0 (overestimate) ○ F’’ > 0 (underestimate) Inverse Functions ● Make sure the function is one-to-one ● You will typically either be given a function and an xvalue (where you’re expected to find the y-value) or a point. ● Often you will have to know that a point on the inverse of a function is opposite on the original function ○ Ex: (1,2) ---> (2,1) on inverse ● Use the theorem: ○ (F-1)’ (a) = 1/f’(g(x))

Integration: ● Integration=area under the curve ● Definite integrals are defined from a to b (integrals with numbers on top and bottom). ● Indefinite integrals are not defined on an interval. ● Riemann Sums are approximations of definite integrals: ○ When doing Riemann Sums: i. Construct rectangles using the number of intervals given, intervals may not be equal. ii. On the bottom, put x-values and if needed the interval (changes) in circles. iii. On the top of each rectangle put f(x) or the corresponding y-value. iv. From there you can use any riemann sum method to solve: left hand rule (exclude last term), right hand rule (exclude first term), midpoint formula (construct midpoints to use), or the trapezoidal rule (double middle terms or take separate trapezoids). v. You may have to separate figures into separate terms to multiply them by a different base then add them together. vi. Use corresponding formula (b*h,0.5h(b1+b2), etc) to form your riemann sum and solve. vii. Only use the numbers you need (sometimes tables are given with extra numbers but don’t

be fooled). viii. Make sure you only use numbers at the ends/beginnings/middles of each rectangle. ix. When given a riemann sum that can be done with a calculator DON’T stray away from the riemann sum method. x. Some riemann problems (especially word problems) will require an extra step (such as subtracting your answer from the original number given). xi. ○ You may have to determine if what you found is an overestimate or underestimate in which case you: i. See if the rectangles are inscribed (underestimate) or not (overestimate). ii. Justify by saying what riemann sum you used and whether the graph is concave up or down, increasing or decreasing . ● Displacement (area under curve): i. Just the integral of change in y. ii. When you have a definite integral either look at the function and figure out what the graph would look like. Then use geometric formulas to solve for the area. iii. OR plug in the defined values into the function and subtract one from the other

● s(b)-s(a) ● Know all of these theorems for integrals:

○ And These:

○ What to do with powers: ■ The integral of number*ex is that number*ex

● Integral of 5ex = 5ex ■ Powers that are multiplied (as long as the base is the same) can be added. ■ Powers that are divided (as long as the base is the same) can be subtracted. ■ If you see a function in parenthesis to a power then distribute that power (as long as it’s easy otherwise use u-substitution). ● Solving Differential Equations: ○ Given dy/dx = f(x) ○ Cross multiply ○ Take the integral of both sides to get rid of d in dy ○ You should have a general solution and DON’T FORGET TO ADD “C” TO THE END as this is a general solution ○ ANYTIME YOU HAVE A GENERAL SOLUTION OF AN INDEFINITE INTEGRAL ALWAYS ADD “C”. ○ If you were given a point at the beginning of the problem use the y-value as y and the x-value as x in the function to solve for “C”. ○ Now substitute that value in for “C” in the equation and you have found THE particular solution.

● Slope Fields: ○ Graphs of slopes ○ What do I do? ● You can integrate with a visual graph: ○ Use a geometrical formula to find the area of the shape formed from the definite integral. ○ Don’t forget that area above the x-axis is positive and area below the x-axis is negative. THIS REMAINS TRUE WHEN ADDING THE AREAS TOGETHER. ● Adding/Subtracting/Working with multiple integrals: ○ Integrals will be given ○ Be careful here ○ Draw a number line ○ Check to see if the given integrals have their numbers in the right spots (a,b). ○ If not switch them and multiply by a negative ○ THEN make a problem (addition or subtraction usually) that you can solve ○ Alternatively you can add the two smaller bounds to equal the largest bound and solve from there. ○ Watch out since that answer may be multiplied/divided/modified in some way. ● Using the Calculator to find integrals:

○ Graph: ■ Enter the function in the “y=” section (where you enter functions to be graphed). ■ Go to the “graph” tab ■ Hit “2nd” and “trace” followed by “7” ■ Type in the applicable x-values followed by hitting enter (a,b). ■ Done. ○ Regular: ■ Hit “math”, “9” ■ Specify “a” and “b” ■ Enter the function ■ Enter “X” after d ■ Hit enter ■ The Calculator with solve for a number. ○ AND USE THE CALC AS MUCH AS YOU CAN ■ To get absolute value signs - “math” “>” “abs.” ● Integrating rational functions: ○ Separate the numerator into pieces that can be placed over the denominator (only works if the denominator is one term). ○ Many times you can bring the denomina...