Calculus 1 Overview
Reimann Sum
"Approximating the area under a curve." Area = ΣΔxf(x) Δx = b-a/n Left: underestimate Right: overestimate Midpoint: closest accuracy Useful formulas: 1. ΣC = nC 2. Σk = (n(n+1))/2 3. Σk^2 = (n(n+1)(2n+1))/6q Limits: ∫f(x)dx = lim x→∞ ΣΔx•f(xi) xi = a+Δx•i
Derivative of bˣ
(bˣ) • ln(b)
Derivative of arccsc(x)
-1/|x|(√(x²-1))
Derivative of arccos(x)
-1/√(1-x²)
Derivative of csc(x)
-csc(x)cot(x)
Derivative of cot(x)
-csc²(x)
Derivative of cos(x)
-sin(x)
In(determinate) Forms
0/0, ∞/∞, ∞*0, ∞ - ∞, 1^∞, 0⁰, ∞⁰ -- "Do more work!" #/0, 0/#, # -- "Almost done!"
Product Rule
1'2 • 2'1
Limits
1. check form 2. use L'Hopital's Rule 3. use L'Hopital's Rule upside down 4. divide by highest power 5. conjugate 6. ? algebra ?
Optimization
1. draw and label visual 2. determine if f(x) is maximized or minimized 3. find constraints 4. solve for a single variable 5. use calculus to optimize problems
Related Rates
1. introduce variables, identify given rates, account for unknowns 2. draw a picture 3. find known equations to solve problem 4. implicitly differentiate with respect to y 5. evaluate what is needed and solve
Implicit Differentiation
1. replace f(x) with y 2. differentiate y on both sides by using dy/dx 3. use chain rule when differentiation y 4. solve for y'
Derivative of Complex Functions
1. replace f(x) with y 2.take natural log of both sides 3. use log rules of differentiation to simplify 4. solve for y' 5. replace y with original equation
Derivative of arccot(x)
1/(1+x²)
Derivative of arctan(x)
1/(1-x²)
(d/dx)(f⁻¹(x)) at x=b
1/(f'(f⁻¹(x)) -or- 1/f'(a) where b=f(a)
Derivative of logb(x)
1/(x(ln(b))
Derivative of arcsec(x)
1/|x|(√(x²-1))
Derivative of arcsin(x)
1/√(1-x²)
First and Second Fundamental Theorem of Calculus
A(x) = ∫f(t)dt on [a,x] -- accumulation function ∫f(x)dx on [a,b] = F(b)-F(a) if F'(x)=f(x)
Formulas for the Pyramid
B = area of base V = (1/3)Bh
Formulas for the Cone
B = πr² V = (1/3)πr²h
Continuity and the Intermediate Value Theorem
Continuous = Differentiable (smooth continuous) Discontinuity: holes, VA, NA, unequal limits, corners, cusps IVT: between f(a) and f(b) on the inteval [a, b], f(c)=d -- check by plugging in each interval endpoint interval into the original function to find where it equals zero
Distance Formula
D = √((a-x)² + (b-y)²)
Antiderivatives and Area
Displacement: ∫v(t)dt , [a,b] -- s(b)-s(a) Distance Travelled: ∫|v(t)|dt , [a,b] -- 2s(c)-s(a)-s(b)
Antiderivatives
F'(x)=f(x)
Mean Value Theorum
If F(x) is CONTINUOUS on [a,b] and DIFFERENTIABLE on (a,b), then there is a c in (a,b) that F'(c)=[F(b)-F(a)]/(b-a).
L'Hopital's Rule
If the limx→c (f(x)/g(x)) is indeterminate, then the limx→c (f(x)/g(x))=limx→c(f'(x)/g'(x)). Use to find limits of inderminate forms by taking the derivative of the top and bottom.
Linear Approximation
L(x) = f(a) + f'(a)(x-a)
Formulas for the Rectangle
P = 2a + 2b A = ab
Formulas for the Circle
P = 2πr A = πr²
Formulas for the Triangle
P = a + b + c A = (1/2)bh
Formulas for the Trapezoid
P = a + b + c + d A = ((a+c)/2) • h
Substitution
Use 'u' in place of a complicated piece of the function when computing an integral.
Formulas for the Sphere
V = (4/3)πr³ SA = 4πr²
Formulas for the Cube/Rectangular Box
V = lwh or a³ SA = 2lw + 2wh + 2lh or 6a²
Formulas for the Cylinder
V = πr²h SA = 2πr² + πr • h
Graphing Basics
VA: when x=0 in the denomonator HA: the highest powers between the numerator and the denomonator y-Intercepts: when y=0 for the function x-Intercepts: when x=0 in the numerator Sign Charts: first derivative and second derivative for points of interest ***end points do not matter***
Understanding Functions
When given f(#), answer with a value of the y-axis. When given a limit where x→#, answer with a value of +∞ or -∞. Domains = unions Intervals = commas Even: f(-x)=f(x) Odd: f(-x)=-f(x)
Quotient Rule
[(lo)(hi')-(hi)(lo')]/lo²
Derivative of sin(x)
cos(x)
Chain Rule
d/dx f(g(x)) = f'(g(x)) • g'(x)
Velocity
derivative of speed or position (first derivative)
Acceleration
derivative of velocity (second derivative)
Logarithmic Differentiation
dy/dx = (f(x)ᵍ⁽ˣ⁾) • (g'(x) • ln(f(x)) + g(x) • (f'(x)/f(x))
Derivative of eˣ
eˣ
Graphing with the First Derivative
f' > 0 — increasing f' < 0 — decreasing f' = 0 or DNE — critical points (i.e. cusps, corners, asymptotes, holes)
Graphing with the Second Derivative
f'' is positive — convave up f'' is negative — concave down f'' = 0 or DNE — point of inflection is marked by a change in concavity and/or f'' changes signs (no POI at discontinuity) → find local extremum
Intergral Application
f(avg) on [a,b] = (∫f(t)dt)/(b-a)
Graphing with the First and Second Derivative
f(x) → CCD, increasing - f'(x) is increasing - f''(x) is negative f(x) → CCD, decreasing - f'(x) is decreasing - f''(x) is negative f(x) → CCU, increasing - f'(x) is increasing - f''(x) is positive f(x) → CCU, decreasing - f'(x) is decreasing - f''(x) is positive
First Definition of a Derivative
f(x)-f(a)/x-a 1) Δy/Δx — slope of a secant line through (x, f(x)) and (a, f(a)) 2) Δposition/Δtime ⇔ (s(b)-s(a))/(b-a) — average velocity between the time x and a, if f is a position function 3) Δvalue/Δinput ⇔ (f(b)-f(a))/(b-a) — average rate of change for f between x and a
Squeeze Theorum
if f(x)<g(x)<h(x) when x is near A (but not A) and limf(x)=limh(x)=L then limg(x)=L.
Second Definition of a Derivative
lim h→0 for f(x+h)-f(x)/h
First Definition of a Derivative (Limits)
lim x→a for f(x)-f(a)/x-a 1) y=f(x) at x=a — slope of a tangent line 2) instantaneous velocity at time a if f is a position 3) lim (average rate of change) — instantaneous rate of change of f at a
Derivative of xⁿ
nxⁿ⁻¹
Derivative of sec(x)
sec(x)tan(x)
Derivative of tan(x)
sec²(x)
Trig Identites
tan(θ) = sin(θ)/cos(θ) cot(θ) = cos(θ)/sin(θ) sec(θ) = 1/cos(θ) csc(θ) = 1/sin(θ) cos²(θ) + sin²(θ) = 1 1 + tan²(θ) = sec²(θ) a² + b² = c²
Slope of a Tangent Line
y-y₁=b(x-x₁)
Converting Degrees to Radians
θ° • π/180° = x radians
Definite Intergrals
∫f(x)dx , [a,b]
Indefinite Integrals
∫f(x)dx = F(x)+c Initial Value Problem