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89 Cards in this Set
- Front
- Back
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Slope of Tangent
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Limit of the slopes of the secant lines as Q gets closer to P
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lim f(x) as x approaches a = L
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If we can make the values of f(x) arbitrarily close to L by taking x to be sufficiently close but not equal to a
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Vertical Asymptote
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Either the lim of f(x) on one or both sides approaches infinity or negative infinity
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lim (f(x))^n as x approaches a
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(lim f(x) as x approaches a)^n
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Root Rule lim nth root of f(x) as x approaches a
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nth root of the limit of f(x) as x approaches a IF n is even and f(x) > 0
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DS (Direct Substitution)
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If f(x) is a polynomial or a rational function and a is in the domain of f then lim f(x) as x approaches a = f(a)
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The Squeeze Theorem
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If f(x) < g(x) < h(x) near a and lim f(x) as x approaches a = lim h(x) as x approaches a = L then lim g(x) as x approaches a = L
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Continuous
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lim f(x) as x approaches a = f(a) a belongs to the domain of f and lim f(x) as x approaches a exists
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Removable Discontinuity
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Hole in the graph due to not defined at point or defined at a number not equal to the limit
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Jump discontinuity
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Jump in the graph of f
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Infinite Discontinuity
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The limit approaches infinity from one or both sides at a
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Greatest Integer Function
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[|x|] = the largest integer less than or equal to x
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Df = [d,b]
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f continuous at a belongs to (d,b)
f cont. from right at d f cont. from left at b |
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Functions Continuous on their domains
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-Polynomials
-Rational Functions -Root Functions -Trigonometric Functions |
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Limit definition of Derivative
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f'(x) = lim [f(x+h) - f(x)]/h as h approaches 0
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Differentiable
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Differentiable at a if f'(a) exists
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Failing Differentiability
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-Corner or Kink
-Not Continuous -Vertical Tangent |
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d/dx[x^n]
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nx^(n-1)
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d/dx[f(x)*g(x)]
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f'(x)*g(x) + g'(x)*f(x)
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d/dx[f(x)/g(x)]
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[g(x)*f'(x) - f(x)*g'(x)]/[g(x)]^2
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Infinite Discontinuity
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The limit approaches infinity from one or both sides at a
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Greatest Integer Function
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[|x|] = the largest integer less than or equal to x
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Df = [d,b]
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f continuous at a belongs to (d,b)
f cont. from right at d f cont. from left at b |
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Functions Continuous on their domains
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-Polynomials
-Rational Functions -Root Functions -Trigonometric Functions |
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Limit definition of Derivative
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f'(x) = lim [f(x+h) - f(x)]/h as h approaches 0
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Differentiable
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Differentiable at a if f'(a) exists
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Failing Differentiability
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-Corner or Kink
-Not Continuous -Vertical Tangent |
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d/dx[x^n]
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nx^(n-1)
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d/dx[f(x)*g(x)]
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f'(x)*g(x) + g'(x)*f(x)
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d/dx[f(x)/g(x)]
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[g(x)*f'(x) - f(x)*g'(x)]/[g(x)]^2
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Horizontal Tangent
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f'(x) = 0
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Derivate of sin(x)
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cos(x)
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Derivative of cos(x)
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-sin(x)
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Derivative of tan(x)
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sec^2(x)
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Derivative of cot(x)
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-csc^2(x)
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Derivative of sec(x)
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sec(x)tan(x)
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Derivative of csc(x)
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-csc(x)cot(x)
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lim sinx/x as x approaches 0
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1
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Chain Rule
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d/dx[f(g(x))] = f'(g(x))*g'(x)
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Implicit Differentiation
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Differentiate both sides and solve for the correct variable (usually y')
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sin^2(x) + cos^2(x)
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1
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tan^2(x) + 1
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sec^2(x)
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1 + cot^2(x)
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csc^2(x)
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sin(2x)
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2sin(x)cos(x)
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cos(2x)
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cos^2(x) - sin^2(x)
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cos^2(x)
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(1 + cos(2x))/2
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sin^2(x)
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(1 - cos(2x))/2
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Linear Approximation: L(x)
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y = f(a) + f'(a)(x-a)
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Absolute Maximum and Minimum
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Max - f(c) greater than or equal to f(x) for all x in D
Min - f(c) less than or equal to f(x) for all x in D |
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Extreme Value Theorem (EVT)
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If f cont. on [a,b] then it attains abs. max f(c) and abs. min f(d) at some numbers c and d in [a,b]
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Fermat's Theorem
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If f has a local extreme value and f'(c) exists then f'(c) = 0
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Critical Number
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A number c in the domain of f such that either f'(c) = 0 or f'(c) D.N.E.
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Closed Interval Method
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Find values of f at critical numbers c in [a,b], find f(a) and f(b). The largest number is abs. max and smallest is abs. min.
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Rolle's Theorem
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f is cont. on [a,b], f is diff. on (a,b), f(a) = f(b) then there is a c in (a,b) such that f'(c) = 0
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Intermediate Value Theorem (IVT)
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f cont. on [c,d], f(c)<0, f(d)>0 then there exists some number a between c and d such that f(a) equals 0.
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Mean Value Theorem (MVT)
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f is cont. on [a,b], f is diff. on (a,b) then there is a c in (a,b) such that f'(c) = [f(b)-f(a)]/(b-a)
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If f' is positive
If f' is negative |
f is increasing
f is decreasing |
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First Derivative Test:
f' switches from positive to negative or negative to positive If f' doesn't switch at c |
Local max or local min
If f' doesn't switch at c |
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The Concavity Test:
If f''(x)>0 for all x in I If f''(x)<0 for all x in I |
f(x) is CU on I
f(x) is CD on I |
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Second Derivative Test:
If f'(c)=0 and f''(c)<(>)0 |
f has local max(min) at c
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Horizontal Asymptote: L
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lim f(x) as x approaches infinity or negative infinity equals L
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If r>0 is a rational number then lim 1/(x^r) as x approaches infinity =
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0
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If top and bottom of infinite limit are same power then
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Horizontal asymptote at top coefficient of highest power / bottom coefficient of highest power
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If top power is lower than bottom power of infinite limit then
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lim equals 0
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Optimization
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PRACTICE
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Antiderivative: F
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If F'(x) = f(x) for all f in I
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General Antiderivative
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F(x) + C
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Antidifferentiation Formulas
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Anti of non co trig functions NEGATIVE
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Area
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Region enclosed by certain curves
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Riemann Sum
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Integral of f(x)dx = lim as n approaches infinity of the sum from i=1 to n of f(xi*) times delta x
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Integral from a to b of f(x)dx
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lim as n approaches ifinity of the sum from i=1 to n of f(xi) times delta x
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Delta x
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(b-a)/n
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xi
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a + i * delta x
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Fundamental Theorem of Calculus part 1:
If f is cont. on [a,b] then g(x) is defined by: g(x) = integral from a to x of f(t)dt where a is less than or equal to x is less than or equal to b is cont. on [a,b] and diff. on (a,b) and g'(x) = f(x) OR |
d/dx[the integral from a to x of f(t)dt] = f(x) = f(upper bound) d/dx(upper bound)
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Fundamental Theorem of Calculus part 2: If f cont. on [a,b]
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then the integral from a to b of f(x)dx = F(b) - F(a) where F is any antiderivative of f
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If f is not cont. on [a,b] for the integral from a to b of f(x)dx
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Then the FTC CANNOT be used
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The indefinite integral of f(x)dx =
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F(x) + C
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Integral of x^n =
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(1/(n+1)) * x^(n+1) + C
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Substitution Rule
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PRACTICE
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Area Between two curves
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Integral from a to b of upper curve minus lower curve IF f(x) and g(x) are greater than or equal to 0
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If two functions are odd and the bounds of the integral are opposites then
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the integral is equal to 0
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The integral in terms of area equals
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Area above x axis minus area below x axis
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Volume of a solid equals
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lim as n approaches infinity of the sum from i=1 to n of A(xi) times delta x = the integral from a to b of A(x)dx
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Discs to find solid of rotation volume
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Integral from a to b of Area of disc times width of disc
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Washers
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Integral from a to b of area of outer disc minus area of inner disc
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Shells (volume of 1 shell)
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V = 2*pi*r*h*delta r where r = .5(r1 + r2) and delta r = r2 - r1
V = circumference * height * thickness |
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Shells (volume of solid)
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Integral from a to b of 2*pi*xf(x)dx where 2*pi*x = circumference, f(x) = height and dx = thickness
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Shells: Rotating about x or y axis
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Use the same variable as the axis of rotation for the integral
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Washers and Discs: Rotating about x or y axis
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Use the opposite variable as the axis of rotation for the integral
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