AP Calculus AB and AP Calculus BC - Popelka (WI)
*Average Value of A Function* (involves integrals) (NOT EQUIVALENT TO AVERAGE RATE OF CHANGE OR MVT)
(1/b-a) * The integral from a to b of f(x) dx Represents the height of the rectangle that has the same area as under the curve.
*Average Velocity* Integral?
(1/b-a) * The integral from a to b v(t) dt
*Average Speed* Integral?
(1/b-a) * The integral from a to b | v(t) | dt
Slope of A Secant
(Sometimes these can be derivatives in disguise)
Derivative of A Single Variable
(d/dx)(x) = 1
Derivative of Natural Log (ln)
(d/dx)[ln u] = u'/u u > 0 u can not be zero
Average Rate of Change (Slope)
(y2 - y1)/(x2 - x1)
*Need an amount? What do we do...*
*INTEGRATE A RATE*
*Volume by Revolution - Washer Method*(The integrals) 1. Horizontal Axis? 2. Vertical Axis?
1. V= pi * The integral from x=A to x=B of (the outer radius^2 - the inner radius^2) dx 2. V= pi * The integral from y=A to y=B of (the outer radius^2 - the inner radius^2) dy
*Method to Differential Equations by Separation of Variables: Steps? *
1. We get the dependent variables on the left and the independent variables on the right 2. We cross multiply to solve for what is on top 3. Integrate both sides 4. Solve for the dependent variable (dh/dt) = k(h^(1/2) ) kdt = dh/(h ^ (1/2) )
Inverse Trig Functions: Radians 1. arcsin (1/2) = ? 2. arccos(0) = ? 3. tan^-1 (1) = ? 4. sin^-1 (-1) = ? 5. cos^-1 (-(3^1/2)/2) = ?
1. arcsin (1/2) = *pi/6* 2. arccos(0) = *pi/2* 3. tan^-1 (1) = *pi/4* 4. sin^-1 (-1) = *-pi/2* 5. cos^-1 (-(3^1/2)/2) = *5pi/6*
Thinking in Theorems: Rolle's Theorem
Let f be *continuous* on [a,b] *and differentiable* on (a,b). If f(a) = f(b) then there is at least one number (c) in the interval (a,b) such that f'(c) = 0
Derivative of A Constant
1. f(x) = 1 2. f'(x) = 0
Power Rule w/Derivatives of Polynomials
1. f(x)= x^n, n != 0 (n can not equal zero) 2. f'(x) = nx^(n-1) 3. See picture
Define Integral
A method in which the area under a curve can be determined.
Rectangular Surface Area
S = 2pi * The integral from a to b of (f(x))*(1+[f'(x)]^2)^(1/2) dx
Acceleration = ?
a(t) = v'(t) = s''(t)
fa2
af
afa
afa
Horizontal Asymptote
condition: lim f(x) as x approaches + or - infinity = a conclusion: HA: y = a
Vertical Asymptote
condition: lim f(x) as x approaches a = + or - infinity conclusion: VA: x = a
dad
dad
Position/Displacement = ?
denoted by x(t) or something similar
*Definition of A Derivative*
f'(x) = the limit of h as h approaches zero of f(x+h)-f(x)/h (this is sometimes written with delta to signify change - see picture)
Linearization - Justification (Tangent Lines)
f(x) = L(x) for x near x = a *Second Derivative* 1. UP - f(x) is concave up, the tangent line is below the graph of f(x) and the approx. will *underestimate* the actual value 2. DOWN - f(x) is concave down, the tangent line is above the graph of f(x) and the approx. will *overestimate* the actual value
*Fundamental Theorem of Calculus: Part 2* (Ex: g(x) = The integral of 1 to x^2 f(t) dt. What is the derivative of g'(x)?)
f(x^2) * d/dx (x^2) - Chain Rule f(x^2) * 2x
afewf
gf
*The Circle of Derivatives and Integrals - Sine and Cosine*
Derivatives: rotate clockwise (to the right) sin -cos cos -sin Integrals: rotate counterclockwise (to the left) sin -cos cos -sin
Velocity = ?
v(t) (velocity) = s'(t) (speed)
Equation of A Line
y - y1 = m(x - x1)
*Derivatives: Product Rule (Ex: y = f(x)g(x) )*
y' = (f(x))*(g'(x)) + (g(x)) * (f'(x)) *Chant* "The first times the derivative of the second, plus the second times the derivative of the first."
*Derivatives: Quotient Rule (Ex: y = f(x)/g(x) )*
y' = (g(x) * f'(x)) - (f(x) * g'(x))/ (g(x))^2\ *Chant* "Low "d" high minus high "d" low, over low^2"
Derivative of A Power or Exponent (Ex: y = a^u)
y' = (ln a ) a^u * u'
*Method to Differential Equations by Separation of Variables Continued : Special Circumstance* Ex: dy/dx = ky/1 y=?
y=Ce^kx (This one is just good to know)
*Integration Method: Partial Fractions
Factoring is key here! This method usually only turns up when you have a linear equation over a polynomial equation. Separate the polynomial fraction into different parts. Now, multiply diagonally (butterfly method) to obtain the following (ex): A(3x +2) + B(x+5) = -3x +24 Now, you will need to find an x value that will "cancel out" A and make it 0 (such as -2/3). The same will need to be done with B. Each time you will obtain an answer for either A or B. Going back to the integral, we plug in our values for A and B. Then, we integrate. Usually you will get two natural logs, in which case you can simplify using the properties of logs.
Thinking in Theorems: Mean Value Theorem
If f is cont. on [a,b] and differentiable on (a,b), then there exists a number where f'(c) = f(b) - f(a)/b-a (where the derivative at a certain point equals the slope of the secant line)
Thinking in Theorems: Intermediate Value Theorem
If f(x) is a *continuous* function on [a,b], then f(x) will take on every value between f(a) and f(b) at least once (every point is being taken on between the interval f(a) and f(b) )
The Integrals of Trig Functions *Diff.* 1. d/dx[sinx] = cosx Now determine the integrals! 2. d/dx[cosx] = -sinx 1. cosx 3. d/dx[tanx] = sec^2 x 2. sinx 3. sec^2x
*Int.* 1. The integral of cosx dx = sinx + C 2. The integral of sinx dx = -cosx + C 3. The integral of sec^2 x dx = sec^2 x + C
Indefinite Integration *Diff. * 1. d/dx[kx] = k Now determine the integrals! 2. d/dx[x^n] = nx^(n-1)
*Int.* 1.The integral of k dx = kx+c 2. The integral of x^n = (x^(n+1)/n+1) + C (Can't forget that c!!) n != -1 (n does not equal -1)
*Integration Method: Integration by Parts - U Determination Order?*
*L*og *I*nteger *A*algebraic Expression *T*rigonometric Function *E*xponents
Introduction to Integration (Ex: 3x^2)
*Method:* 1. The integral of 3x^2 dx 2. Bring the constant out front (the 3) 3. Add one to the power and divide by the new power (3) 4. Add the constant "C" 5. Now, we can determine that the integral of 3x^2 is x^3 + C
*Special Integration: tanx*
*Note: The answer is NOT sec^2 x + C* 1. The integral of tanx dx = The integral of sinx/cosx dx 2. U-sub cosx and get the du of cosx 3. We now have the - integral of 1/u du = -ln|u| + C 4. Put back in our u (cosx) to get the following: *-ln|cosx| + C
*Function Continuity Rules* (common with piece-wise functions)
*The function is continuous.... 1. If the lim f(x) as x approaches at some constant (c) 2. f(c) exists 3. The limit of f(x) as x approaches c equals f(c) ALL RULES must be met in order for the function to be continuous
(x^2)^(1/2) = ?
*|x|* NOT x
Derivatives: Inverse Trig Functions 1. (d/dx) (sin^-1 u) = ? 2. (d/dx) (cos^-1 u) = ? 3. (d/dx) (tan^-1 u) = ? Sometimes they are written as: arcsin arccos arctan
1. (d/dx) (sin^-1 u) = *1/(1-u^2)^(1/2) * (du/dx)* 2. (d/dx) (cos^-1 u) = *-1/(1-u^2)^(1/2) * (du/dx)* 3. (d/dx) (tan^-1 u) = 1/1+u^2 * (du/dx)
Derivative of A Circle (Ex: x^2 + y^2 = 25)
1. (d/dx)(x^2) + (d/dy)(y^2) = (dy/dx) 25 2. 2x + 2y * y' = 0 - *Implicit Differentiation* 3. y' = -2x/2y 4. *y' = -x/y*
*1. When is an object speeding up? 2. When is an object slowing down?*
1. An object is speeding up when the acceleration and the velocity of the object are the same sign. 2. An object is slowing down when the acceleration and the velocity of the object are NOT the same sign. Ex: a(t) = -9 vs. v(t) = 22 The particle or object is slowing down then.
Thinking in Theorems: Mean Value Theorem Steps
1. Calculate the slope of the secant line (mSec) 2. Find the derivative (dy/dx) of the function 3. At some value of x, the slope of the secant line is *equal to* the derivative of the function 4. Now, make sure your answer is in the given interval (if not omit or start over to find mistakes)
Integrals of Inverse Trig Functions 1. The integral of du/(a^2 - u^2)^(1/2) = ? 2. The integral of -du/(a^2 - u^2)^(1/2) = ? 3. The integral of ln u du = ? (Hint: take the square root of a^2 and u^2 to get the a and u for the integrals) *Note: If there is an x near the du DO NOT USE INVERSE TRIG FUNCTIONS (especially tan^-1). It is most likely an ln case.*
1. The integral of du/(a^2 - u^2)^(1/2) = *sin^-1 + C* 2. The integral of -du/(a^2 - u^2)^(1/2) = *1/a tan^-1 - (u/a) + C 3. The integral of ln u du = *u ln u - u + C
*Fundamental Theorem of Calculus: Part 1* 1. The integral from a to b of f(x) dx = ? 2. What is the derivative F'(x) ?
1. F(b) - F(a) 2. f(x) (Might need to fix this card)
The Second Derivative Test
1. If f''(x) > 0, then f(x) has a relative min at (x, f(x)) (f(x) is facing up) 2. If f''(x) < 0, then f(x) has a relative max at (x, f(x)) (f(x) is facing downward) 3. If f''(x) = 0, then the first part of the test fails and you have to use the First Derivative Test
The First Derivative Test ( Relative Min and Max)
1. If f'(x) changes from (-) to (+) at c then f has a relative minimum at (x,f(x)) 2. If f'(x) changes from (+) to (-) at c then f has a relative max at (x, f(x))
*Integration Method: U-Substitution*
1. Pick "u" 2. Find "du" and adjust it to match the "dx" term 3. Replace the integral w/"u" and "du" terms 4. Integrate 5. Replace answer w/"x" terms *and add C*
When do you know an object is going to the right or left when given the acceleration?
1. Pushed Right: a > 0 (or moving up) 2. Pushed Left: a < 0 (or moving down)
When do you know an object is going to the right or left when given the velocity?
1. Right: v > 0 (or moving up) 2. Left: v < 0 (or moving down)
Displacement ( What is the definition and integral?)
1. The difference in distance from where you start and where you stop. 2. The integral from a to b of v(t) dt
The Indefinite Integral of 1/x
1. The integral of 1/x dx is equivalent to the integral of x^-1 2. The integral of 1/x is ln|x| + C dx
Straight Line Motion Revisted: 1. v(t) = The integral of what? 2. s(t) = The integral of what?
1. v(t) = The integral of a(t) dt 2. s(t) = The integral of v(t) dt
*Implicit Differentiation: Derivative of xy*
1. xy' + y' (1) Implicit w/Product Rule (applies to most Implicit Differentiation problems)
*Derivative of A Square* (Ex: y= (x)^(1/2) )
1. y = (x)^(1/2) 2. y' = (1/2)*x^(-1/2) 3. y' = 1/2(x^1/2)
Derivative of A Fraction (Ex: y = 1/x)
1. y = 1/x = x^-1 2. y' = (-1)(x^-2) = -1/x^2 3. y' = -c/x^2 (c is a constant)
Derivative of Trig Functions 1. y = sinx 2. y = cosx 3. y = tanx
1. y' = cosx 2.y'= -sinx 3. y' = sec^2 (x)
*Implicit Differentiation: Derivative of y^2
2y * y'
When do you use u-substitution?
Most of the time when the derivative can be found in the integral and "cancel out" any one of the terms in the integral.
Average Rate of Change
Msec = f(x+h) - f(x)/h ( known as the Difference Quotient)
Arc Length - What is the Formula?
The integral from a to b of (1 + [f'(x)])^2)^(1/2)
Distance (not to be confused with displacement. What is the integral?)
The integral from a to b of | v(t) | dt
*Special Integration: 1/x*
The integral of 1/x dx = ln|x| + C
*Special Integration: e^x*
The integral of e^x dx = e^x + C (This is unless if there is a number next to the x) Ex: e^(3x) When Integrated: (1/3)(e^(3x))
*Integration Method: Integration by Parts* *Chant:* "ultraviolet voodoo"
The integral of u dv = uv - the integral of v du u = --> v = (then go to du) du = dv=
Approximation Ace: Euler's Method
The numerical method of solving differential equations by calculating approximate y-values. Usually given a step size or a range of values to divide into equal step sizes. Note: delta y/delta x = m (slope) and Ynew = (dy/dx)the change in x + Yold
L'Hospital Rule
This is usually when the limit is infinity over infinity or 0/0. It can be any indeterminate form. 1 ^ infinity is another example. (This is NOT the Quotient Rule)
*The Area Between Two Curves* (Incorporates the Fundamentals of Integration) f(x) and g(x)
Total Area: The integral from a to b of [f(x) - g(x)] dx (Which ever one is on top is the "bigger function" to be subtracted from)
*Volume by Revolution -Disc Method*
V= pi * The integral from a to b of (R(x))^2 (or the radius squared) dx
If a function can be derived, then what else can we determine?
We can determine that the function is also continuous. Diff. --> Cont.
*Derivatives: Chain Rule*
We multiply by the derivative of the "guts"
*Integration Method: Integration by Parts - "DIS" method or Tabular Method (can only be used in certain circumstances and most of the time can't be used with natural log problems)*
We take the u and get the derivative until it reaches 0. Then, we take the integral of v until we surpass the point where u reaches 0 by 1 block. Finally, we multiply in a *diagonal direction* and apply "ultraviolet voodoo" to the problem at hand. (u) (v) | Derivative | Integral | Sign | | | | + | | | | - | | | | + | (+, -, +, -, etc.)
When do you use the Partial Fraction Method when integrating?
When there is a line function on top of the fraction and there is a parabola on the bottom. (Double check to make sure it is not something else.)
