definitions Flashcards
Intermediate Value Theorem (IVT)
If f continuous on [a,b], the f takes on all values between f(a) and f(b)
Average Rate of Change on the Interval [a,b]
AROC = f(b)-f(a) / b-a
Definition of a Vertical Asymptote
x = a is a VA of f if lim x->a f(x) = +/- ∞
Definition of a Horizontal Asymptote
y = b is a HA of f if lim x->+/- ∞ f(x) = b
Squeeze Theorem
Let g(x) ≤ f(x) ≤ h(x) for x ≠ a on some interval about x = a. If lim x->a g(x) = lim x->a h(x) = L, then lim x->a f(x) = L
Definition of Continuity at a Point
If lim x->a f(x) = f(a), the f is continuous at x = a
Existence of the Limit of a function
lim x->a f(x) = L if and only if lim x->a- f(x) = lim x->a+ f(x) = L
Limit Definition of the Derivative
f’(x) = lim h->0 f(x+h)-f(x) / h
Alternate Definition of the Derivative
f’(x) = lim x->a f(x) - f(a) / x - a
Mean Value Theorem of Derivative
If f continuous on [a,b] and differentiable on (a,b) then there exists a c in (a,b) at which f’(c) = f(b) - f(a) / b - a
Extreme Value Theorem
If f is continuous on [a,b] then f has both a min and max on [a,b]
Interior Extremum Theorem
If f is differentiable on (a,b) and f has a local max/min on c (a,b), the f’(c) = 0
Monotonic Function
A function is monotonic on an interval if it is always increasing or decreasing on the interval
Concavity Definition
f is concave up where f’ is increasing and f” > 0
f is concave down where f’ is decreasing and f”<0
Inflection Point Definition
A point in the domain of f where there is a tangent line and concanvity changes
Critical Point definition
A point at which f’ = 0 or f’ DNE
1st Derivative Test
If f’ changes + to - at x = c, then f has a local max at x = c
If f’ changes - to + at x = c, then f has a local min at x = c
If f’ does not change sign, there is neither a min nor a max
2nd Derivative Test
If f’ = 0 and f” > 0 at x = c, then f has a local min at x = c
If f’ = 0 and f” < 0 at x = c, then f has a local max at x = c
Points at which f is not differentiable (4)
Corner, Cusp, Jump, Infinite discontinuity
Stationary Point
A point at which f’(x) = 0
L’Hospital’s Rule
If lim x->a f(x) / g(x) results in 0 / 0 or ∞ / ∞ then lim x->a f(x) / g(x) = lim x->a f’(x) / g’(x)
Fundamental Theorem of Calculus Parts 1 and 2
the integral from a to b of f(x) dx = F(b) - F(a)
the derivative of the integral from a to x of f(t) dt = f(x)
Mean Value Theorem for Integrals
If f is continuous on [a,b] then at some point c in [a,b], f(c) = 1 / b-a of the integral a to b of f(x) dx
Average Value of a Function
1 / b - a from the integral of a to b of f(x) dx
Trapezoidal Rule
integral from a to b of f(x) dx ≈ h / 2 ( f(a) + 2 f(x₁) + 2 f(x₂) + … + 2 f(xₙ) + f(b) `)
Volume
π integral a to b of ( f(x) )² dx
π integral a to b of [ ( f(x) )² - ( g(x) )² ] dx
