AP Calculus AB and AP Calculus BC -
Popelka (WI) QUESTIONS 100%
SOLVED!!
Equation of A Line - ANSWERy - y1 = m(x - x1)
Average Rate of Change (Slope) - ANSWER(y2 - y1)/(x2 - x1)
Vertical Asymptote - ANSWERcondition: lim f(x) as x approaches a = + or - infinity
conclusion: VA: x = a
Horizontal Asymptote - ANSWERcondition: lim f(x) as x approaches + or - infinity = a
conclusion: HA: y = a
Derivative of A Power or Exponent (Ex: y = a^u) - ANSWERy' = (ln a ) a^u * u'
Arc Length - What is the Formula? - ANSWERThe integral from a to b of (1 +
[f'(x)])^2)^(1/2)
(x^2)^(1/2) = ? - ANSWER*|x|* NOT x
Slope of A Secant - ANSWER(Sometimes these can be derivatives in disguise)
Power Rule w/Derivatives of Polynomials - ANSWER1. f(x)= x^n, n != 0 (n can not
equal zero)
2. f'(x) = nx^(n-1)
3. See picture
Linearization - Justification (Tangent Lines) - ANSWERf(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
Average Rate of Change - ANSWERMsec = f(x+h) - f(x)/h ( known as the Difference
Quotient)
*Definition of A Derivative* - ANSWERf'(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)
Derivative of A Constant - ANSWER1. f(x) = 1
2. f'(x) = 0
, Derivative of A Constant - ANSWER
Derivative of A Single Variable - ANSWER(d/dx)(x) = 1
*Derivative of A Square* (Ex: y= (x)^(1/2) ) - ANSWER1. 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) - ANSWER1. y = 1/x = x^-1
2. y' = (-1)(x^-2) = -1/x^2
3. y' = -c/x^2 (c is a constant)
Derivatives Constant Multiple Rule - ANSWER
Derivative of A Circle (Ex: x^2 + y^2 = 25) - ANSWER1. (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*
Rectangular Surface Area - ANSWERS = 2pi * The integral from a to b of (f(x))*(1+
[f'(x)]^2)^(1/2) dx
*Derivatives: Product Rule (Ex: y = f(x)g(x) )* - ANSWERy' = (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) )* - ANSWERy' = (g(x) * f'(x)) - (f(x) *
g'(x))/ (g(x))^2\
*Chant*
"Low "d" high minus high "d" low, over low^2"
Derivative of Trig Functions
1. y = sinx
2. y = cosx
3. y = tanx - ANSWER1. y' = cosx
2.y'= -sinx
3. y' = sec^2 (x)
*The Circle of Derivatives and Integrals - Sine and Cosine* - ANSWERDerivatives:
rotate clockwise (to the right)
sin
-cos cos
-sin
Popelka (WI) QUESTIONS 100%
SOLVED!!
Equation of A Line - ANSWERy - y1 = m(x - x1)
Average Rate of Change (Slope) - ANSWER(y2 - y1)/(x2 - x1)
Vertical Asymptote - ANSWERcondition: lim f(x) as x approaches a = + or - infinity
conclusion: VA: x = a
Horizontal Asymptote - ANSWERcondition: lim f(x) as x approaches + or - infinity = a
conclusion: HA: y = a
Derivative of A Power or Exponent (Ex: y = a^u) - ANSWERy' = (ln a ) a^u * u'
Arc Length - What is the Formula? - ANSWERThe integral from a to b of (1 +
[f'(x)])^2)^(1/2)
(x^2)^(1/2) = ? - ANSWER*|x|* NOT x
Slope of A Secant - ANSWER(Sometimes these can be derivatives in disguise)
Power Rule w/Derivatives of Polynomials - ANSWER1. f(x)= x^n, n != 0 (n can not
equal zero)
2. f'(x) = nx^(n-1)
3. See picture
Linearization - Justification (Tangent Lines) - ANSWERf(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
Average Rate of Change - ANSWERMsec = f(x+h) - f(x)/h ( known as the Difference
Quotient)
*Definition of A Derivative* - ANSWERf'(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)
Derivative of A Constant - ANSWER1. f(x) = 1
2. f'(x) = 0
, Derivative of A Constant - ANSWER
Derivative of A Single Variable - ANSWER(d/dx)(x) = 1
*Derivative of A Square* (Ex: y= (x)^(1/2) ) - ANSWER1. 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) - ANSWER1. y = 1/x = x^-1
2. y' = (-1)(x^-2) = -1/x^2
3. y' = -c/x^2 (c is a constant)
Derivatives Constant Multiple Rule - ANSWER
Derivative of A Circle (Ex: x^2 + y^2 = 25) - ANSWER1. (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*
Rectangular Surface Area - ANSWERS = 2pi * The integral from a to b of (f(x))*(1+
[f'(x)]^2)^(1/2) dx
*Derivatives: Product Rule (Ex: y = f(x)g(x) )* - ANSWERy' = (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) )* - ANSWERy' = (g(x) * f'(x)) - (f(x) *
g'(x))/ (g(x))^2\
*Chant*
"Low "d" high minus high "d" low, over low^2"
Derivative of Trig Functions
1. y = sinx
2. y = cosx
3. y = tanx - ANSWER1. y' = cosx
2.y'= -sinx
3. y' = sec^2 (x)
*The Circle of Derivatives and Integrals - Sine and Cosine* - ANSWERDerivatives:
rotate clockwise (to the right)
sin
-cos cos
-sin
