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Vocabulary
Derivatives
Integrals
Theorems
Miscellaneous
100
f (x) is concave up when…
What is when f ′(x) is increasing or f "(x) > 0
100
What is the derivative of: y = 6x^5 – x + 10
What is y'=30x^4 - 1
100
∫ (x5 + 3x2 – x + 1)dx
What is x^6/6 + x^3 - x^2/2 + x
100
Extreme Value Theorem: If f is continuous on a closed interval, then…
What is… f must have both an absolute maximum and an absolute minimum on the interval.
100
What is the Quotient Rule Song?
What is Lo d Hi - Hi d Lo, down below Lo^2 must go
200
A differential equation is…
What is an equation containing one or more derivatives
200
Calculate the derivative of the function f(x) = e^x(x^2 + 1).
What is f'(x) = e^x(x + 1)^2
200
∫sin(x) dx
What is -cos(x)
200
Fundamental Theorem of Calculus #1
What is if f is an integrable function and g(x) = integral of f(x)dx, then the integral of f(x)dx from a to b = g(b) - g(a)
200
A particle changes direction when…
What is v(t) changes signs.
300
To find extreme values of a function, look for where…
What is f ′ is zero or undefined (critical points).
300
Calculate the derivative of the function: f(x)= 2√x / x+√x
What is f'(x)= -√x / (x+√x)^2
300
∫1/x^3 dx
What is 1/2x^2 +C
300
Mean Value Theorem
What is if f is differentiable for all val's of x in (a, b) and f is continuous at x=a and x=b, then there's at least one number x=c in (a, b) such that f'(c) = [f(b) - f(a)] / b-a
300
If a graph is concave up on a right Riemann sum, is it an overestimate or an underestimate?
What is an overestimate
400
Four ways in which a function can fail to be differentiable at a point
What is • Discontinuity • Corner • Cusp • Vertical tangent line
400
If h(x) = f(g(x)) and f(2) = 5, f'(2) = 6, g(3) = 2 and g'(3) = 5, Find h'(3).
What is 30
400
∫sec(x) tan(x) dx
What is sec(x)
400
Fundamental Theorem of Calculus #2
What is if g(x) = integral of f(t)dt from a to x where a is a constant, then g'(x) = (deriv of x)f(x)
400
If f(x) and g(x) are such that lim f(x) as x --> a =+∞ and lim g(x) as x --> a = 0 then... (A) lim [ f(x) . g(x) ] as x --> a is always equal to 0 (B) lim [ f(x) . g(x) ] as x --> a is never equal to 0 (C) lim [ f(x) . g(x) ] as x --> a may be +∞ or -∞ (D) lim [ f(x) . g(x) ] as x --> a may be equal to a finite value
What is C and D
500
Exponential Growth of Decay: If dy/dt=ky, then...
What is y = Ce^kt , where C is the quantity at t = 0, and k isthe constant of proportionality.
500
Calculate the derivative of the function f(x) = cos(x^3 + x^2 + 1).
What is u = x^3 + x^2 + 1 du/dx = 3x^2 + 2x df/du = -sin(u) df/dx = (df/du)(du/dx)= -(3x^2 + 2x)sin(x^3+ x^2 + 1)
500
Using the rule ∫ sin(kx) dx = −1/kcos(kx) + C with k = 2 we have: ∫ f(x) dx = ∫ cos(2x) dx
What is 1/2sin(2x) + C
500
Intermediate Value Theorem
What is: -if f is continuous for all x in interval [a, b] and y is a number between f(a) and f(b), then there's a number x=c in (a, b) for which f(c)=y -basically, if you have a continuous function and you pick a number on the y-axis in an interval, there's a corresponding x-value in that interval
500
Differntiate: y= tan^3√cot(7x)
What is -21csc^2(7x)(tan^2√cot(7x))(sec^2√cot(7x)) / 2√cot(7x)