Section 1
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Polynomial & Rational Functions
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Section 1
(50 cards)
Polynomial & Rational Functions
c is a real number polynomial { f(x) = ax^n + ax^(n-1) + ... + a0 lim(x→c)[f(x)] = f(c) = ax^n + ax^(n-1) + ... + a0 polynomials: f(x) and g(x) lim(x→c)[f(x)/g(x)] = f(c)/g(c) [g(c)≠0] reference: 2.1
Sandwich Theorem
g(x) ≤ f(x) ≤ h(x) for all x ≠ c in some interval about c and lim(x→c)[g(x)] = lim(x→c)[h(x)] = L then: lim(x→c)[f(x)] = L reference: 2.1
Concavity
Concave up on an open interval I if y' is increasing on I. Concave down on an open interval I if y' is decreasing on I. reference: 5.3
Tangent/Normal to a Curve
y = f(x) P(a, f(a)) slope: m = lim(h→0)[ (f(a + h) - f(a)) / h ] tangent at P is line through P with this slope normal = - 1/slope of the tangent reference: 2.4
Definition of a Derivative
f'(x) = lim(h→0)[ (f(a + h) - f(a)) / h ] AKA: Slope reference: 3.1
Strategy for Solving Max-Min Problems
1. Understand Problem 2. Draw a Model (Pictures!) 3. Graph Function 4. Find Critical Points (f'(x) = 0) and Endpoints 5. Solve Model (Pictures!) 6. Interpret Solution (Make sure it makes sense!) reference: 5.4
Slope
Let P1(X1, Y1) and P2(X2, Y2) be points on a non-vertical line. The slope is: m = rise/run = ΔY/ΔX = (Y2 - Y1)/(X2 - X1) reference: 1.1
Increments
If a particle moves from point (X1, Y1) to the point (X2, Y2), the increments in its coordinates are: ΔX = X2 - X1 ΔY = Y2 - Y1 reference: 1.1
Mean Value Theorem
y = f(x) is continuous at every point of the closed interval [a,b] and differentiable at every point of its interior (a,b), there is at least one point c in (a,b) at which: f'(c) = (f(b) - f(a))/(b - a) reference: 5.2
Implicit Differentiation
... ex: x^2 + y^2 + 2xy = 1 2x + 2y*y' + 2y + 2xy' = 0 x + y*y' + xy' + y =0 y'(y + x) = -(x + y) y' = -(x+y)/(x+y) y' = 1 reference: 4.2
Symmetry
y = f(x) is an: - even function if f(-x) = f(x) - odd function if f(-x) = -f(x) for every x in the function's domain reference: 1.2
Properties of Logarithms
Product Rule: loga(xy) = loga(x) + loga(y) Quotient Rule: loga(x/y) = loga(x) - loga(y) Power Rule: loga(x^y) = yloga(x) reference: 1.5
Absolute Extreme Values
Let f be a function with domain D. Then f(c) is the: - absolute max value on D, iff f(x) ≤ f(c) for all x in D - absolute min value on D, iff f(x) ≥ f(c) for all x in D reference: 5.1
Arc Length
S = r*θ S = arc length r = radius θ = central angle of measure reference: 1.6
Discontinuties
Removable: - can be removed by setting f(0) equal to the limit Jump: - one sided limits exist but have different values Infinite: - ex: e^x Oscillating: - oscillates and has no limit as x→0 reference: 2.3
Area Under a Curve
A = ∫f(x)dx [a,b] = (area above x-axis) - (area below x-axis) reference: 6.2
Differentials
y = f(x) dy/dx = f'(x) dy = f'(x)dx reference: 5.5
Economics-y Calculus (pt. 2)
Max Profit: - r'(x) = c'(x) Min Average Cost: - c(x)/x = c'(x) reference: 5.4
Rules for Differentiation
Constant Value: f(x) = c f'(x) = 0 Power Rule: f(x) = x^n f'(x) = nx^(n-1) Constant Multiple: f(x) = cx^n f'(x) = cnx^(n-1) Sum/Difference Rule: f(x) = u ± v f'(x) = u' ± v' Product Rule: f(x) = uv f'(x) = u'v + uv' Quotient Rule: f(x) = u/v f'(x) = (u'v - uv')/v^2 reference: 3.3
Properties of Limits
L, M, c, and k are real numbers lim(x→c)[f(x)] = L lim(x→c)[g(x)] = M Sum Rule: lim(x→c)[f(x) + g(x)] = L + M Difference Rule: lim(x→c)[f(x) - g(x)] = L - M Product Rule: lim(x→c)[f(x)g(x)] = LM Constant Multiple: lim(x→c)[kf(x)] = kL Quotient Rule: lim(x→c)[f(x)/g(x)] = L/M Power Rule: lim(x→c)[(f(x))^(r/s)] = L^(r/s) [s≠0] reference: 2.1
Derivatives of Inverse Trig Functions
(sinx)^(-1) → 1/(sqrt(1-x^2)) (cosx)^(-1) → -(sinx)^(-1) (tanx)^(-1) → 1/(1+x^2) (cscx)^(-1) → -(secx)^(-1) (secx)^(-1) → 1/(|x|sqrt(x^2-1)) (cotx)^(-1) → -(tanx)^(-1) reference: 4.3
1st Derivative Test for Local Extrema (at endpoints)
At left end point a: - If f' < 0 (f' > 0) for x > a, then f has a local max (min) value at a. At right end point b: - If f' < 0 (f' > 0) for x < b, then f has a local min (max) value at a. reference: 5.3
Increasing/Decreasing Function
Let f be a function defined on an interval I and let X1 and X2 be any two points in I. - increases on I if X1 < X2 ⇒ f(X1) < f(X2) - decreases on I if X1 < X2 ⇒ f(X1) > f(X2) reference: 5.2
Intermediate Value Theorem
If a and b are any two points in an interval on which f is differentiable, the f' takes on every value between f'(a) and f'(b) reference: 3.2
f'(x) Failures
Corner: - ex: f(x) = |x| Cusp: - ex: f(x) = x^(2/3) Vertical Tangent: - ex: f(x): x^(1/3) Discontinuity: - ex: Unit Step Function Differentiability Implies Continuity reference: 3.2
Critical Point
A point in the interior of the domain of a function f at which f' = 0 on f' does not exist reference: 5.1
End Behaviour Model
right end behaviour model: lim(x→∞)[f(x)/g(x)] = 1 left end behaviour model: lim(x→-∞)[f(x)/g(x)] = 1 reference: 2.2
Physics-y Calculus
s(t) [position] s'(t) = v(t) [velocity] s''(t) = v'(t) = a(t) [acceleration] s'''(t) = v''(t) = a'(t) = j(t) [jerk] |v(t)| = speed displacement: object over time interval t → t+Δt Δs = f(t +Δt) - f(t) Free-fall Constants (Earth) - g = 32 ft/sec^2 s = .5*(32)t^2 = 16t^2 (s in feet) - g = 9.8 m/sec^2 s = .5*(9.8)t^2 = 4.9t^2 (s in meters) reference: 3.4
Exponential Growth/Decay
y = k*a^x (k>0) a > 1 = exponential growth 0 < a < 1 = exponential decay reference: 1.3
Rectangular Approximation Method
y = x^2 interval: [0,3] subinterval: .5 LRAM: (.5)(0)^2 + (.5)(.5)^2 + ... + (.5)(2.5)^2 = 6.875 MRAM: (.5)(.25)^2 + (.5)(.75)^2 + ... + (.5)(2.75)^2 = 8.9375 RRAM: (.5)(.5)^2 + (.5)(1)^2 + ... + (.5)(3)^2 = 11.375 LRAM < MRAM < RRAM reference: 6.1
Economics-y Calculus
c(x) = cost of production r(x) = revenue p(x) = r(x) - c(x) = profit MARGINAL anything = find derivative reference: 3.4
Linearization
If f is differentiable at x = a, then the equation of the tangent line: L(x) - f(a) = f'(a)(x-a) OMG! It looks like point-slope! L(x) - f(a) = f'(a)(x - a) y - y1 = m (x - x1) THAT'S AMAZING! reference: 5.5
The Integral of a Constant
∫f(x)dx [a,b] = ∫cdx [a,b] = c(b-a) reference: 6.2
Derivatives of Trig Functions
sinx → cosx cosx → -sinx tanx → (secx)^2 cscx → -cscxcotx secx → secxtanx cotx → -(cscx)^2 reference: 3.5
One-to-One Functions
Function f(x) is one-to-one on a domain D if f(a) ≠ f(b) whenever a ≠ b Does not pass horizontal line test reference: 1.5
Rules for Exponents
a^x * a^y = a^(x + y) a^x / a^y = a^(x - y) (a^x)^y = (a^y)^x = a^(x*y) a^x * b^x = (ab)^x (a/b)^x = a^x/b^x reference: 1.3
Continuity at a Point
Interior Point: - continuous at an interior point of its domain if: lim(x→c)[f(x)] = f(c) Endpoint: - lim(x→a+)[f(x)] = f(a) or - lim(x→b-)[f(x)] = f(b) reference: 2.3
Inflection Point
A point where the graph of a function has a tangent line and where the concavity changes is an inflection point reference: 5.3
Derivative at a Point
f'(a) = lim(x→a)[ (f(x) - f(a)) / (x-a)] reference: 3.1
Extreme Value Theorem
If f is continuous on a closed interval [a,b], then f has both a max and min value on the interval reference: 5.1
2nd Derivative Test for Local Extrema
If f'(c) = 0 and f''(c) < 0, f has local max at x=c If f'(c) = 0 and f''(c) > 0, f has local min at x=c reference: 5.3
Two- Sided Limits
Limit exists if: lim(x→c)[f(x)] = L ⇔ lim(x→c+)[f(x)] = L and lim(x→c-)[f(x)] = L c+ = approaching c from the right c- = approaching c from the left reference: 2.1
Average Rate of Change
[f(t + h) - f(t)]/h AKA: Derivative reference: 2.1
Chain Rule
f(x) = g(h(x)) f'(x) = g'(h(x))*h(x) ex: f(x) = (x^2 + 4)^2 f'(x) = 2(x^2 + 4) (2x) = 4x(x^2 + 4) reference: 4.1
An Integral
∫f(x)dx [a,b] ∫ = integral sign [a,b] = limits of integration f(x) = integrand dx = variable of integration [x] reference: 6.2
Concavity Test
Concave up on any interval where y'' > 0 Concave down on any interval where y'' < 0 reference: 5.3
Local Extreme Values
Let c be an interior point of the domain of f. Then f(c) is a: - local max value at c, iff f(x) ≤ f(c) for all x in some open interval containing c - local min value at c, iff f(x) ≥ f(c) for all x in some open interval containing c reference: 5.1
Exponential/Logarithmic Derivatives
e^x → e^x a^x →a^xlna lnx → 1/x loga(x) → 1/(xlna) reference: 4.4
1st Derivative Test for Local Extrema (at Critical Point c)
At Critical Point c: - If f' changes sign from + → - at c (f' > 0 for x < c and f' < 0 for x > c), then f has a local max value at c. - If f' changes sign from - → + at c (f' < 0 for x < c and f' > 0 for x > c), then f has a local min value at c. - If f' doesn't change sign at c (f' has the same sign on both sides of c), then f has a local extreme value at c. reference: 5.3
Linear Equations
Point-Slope: Y - Y1 = m(X - X1) Slope-Intercept: y = mx + b General: Ax + By = C reference: 1.1
Section 2
(38 cards)