Section 1
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instantaneous rate of change
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Mar 1, 2020
Cards (96)
Section 1
(50 cards)
instantaneous rate of change
f'(a)
derivative of cot(x)
-csc²(x)
∫dx
x+c
∫csc(x)cot(x)dx
-csc(x)+C
Average Rate of Change of f(x) on [a,b]
find: [f(b)- f(a)]/ (b-a)
Fundamental Theorem of Calculus #1
The definite integral of a rate of change is the total change in the original function.
Intermediate Value Theorem
If f is continuous on [a,b] and k is a number between f(a) and f(b), then there exists at least one number c such that f(c)=k
piecewise function is differentiable at x=a where the function splits if
a. f(x) is continuous at x=a b. lim x->a- f'(x) = x->a+ f'(x)
If g(x) is the inverse of f(x), find g'(a)
g'(a) = 1/f'(g(a)) aka the reciprocal of the derivative of f(x) at the corresponding point since on g(x) (a, g(a)) and on f(x) (g(a), a)
Given f'(x): Is f continuous @ c? Is there an inflection point on f @ C?
This is a graph of f'(x). Since f'(C) exists, differentiability implies continuouity, so Yes. Yes f' decreases on X<C so f''<0 f' increases on X>C so f''>0 A point of inflection happens on a sign change at f''
f is continuous at x=c if...
Critical Number
If f'(c)=0 or does not exist, and c is in the domain of f, then c is a critical number. (Derivative is 0 or undefined)
Antiderivative of f(x) from [a,b]
derivative of tan(x)
sec²(x)
Alternative Definition of a Derivative
f '(x) is the limit of the following difference quotient as x approaches c
Mean Value Theorem for integrals or the average value of a functions
First Derivative Test for local extrema
Fundamental Theorem of Calculus #2
Point of inflection at x=k
Horizontal Asymptote
Adding or subtracting antiderivatives
Show that lim x->a f(x) exisits
Show the limit on the left side of a = limit on the righ side of a
Equation of the line tangent at x=a
f'(a)=m point: (a, f(a) do y=mx+b or point-slope
2nd derivative test
If f'(c) = 0 and f"(c)<0, there is a local max on f at x=c. If f'(c) = 0 and f"(c)>0, there is a local min on f at x=c.
Derivative of eⁿ
derivative of sin(x)
cos(x)
f(x) is differentiable if
f(x) must be continuous and smooth (no corners, cusps, undefined, discontinuities)
∫sec²(x)dx
tan(x)+C
∫sin(x)dx
-cos(x)+C
derivative of x
1
∫(1/x)dx
ln|x|+C
L'Hopital's Rule
∫csc²(x)dx
-cot(x)+C
derivative of uvw
uvw'+uv'w+u'vw
Definition of a Derivative
derivative of f'(x)+g(x)
f'(x)+g'(x)
Area between two curves
T(x) = upper B(x) = lower
derivative of x^n
nx^(n-1)
∫(a^u)du
ln(a)*aⁿ+C
Antiderivative of xⁿ
Extreme Value Theorem
If f is continuous on [a,b] then f has an absolute maximum and an absolute minimum on [a,b]. The global extrema occur at critical points in the interval or at endpoints of the interval.
∫sec(x)tan(x)dx
sec(x)+C
derivative of sec(x)
sec(x)tan(x)
Mean Value Theorem
The instantaneous rate of change will equal the mean rate of change somewhere in the interval. Or, the tangent line will be parallel to the secant line.
derivative of cos(x)
-sin(x)
Constants in integrals
Intermediate Value Theorem
If f is continuous on [a,b] and k is a number between f(a) and f(b), then there exists at least one number c such that f(c)=k
∫cos(x)dx
sin(x)+C
derivative of f(g(x))
f'(g(x))g'(x)
Derivative of ln(u)
Section 2
(46 cards)