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Calculus 1
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Terms in this set (31)
Points of Inflection
If (c, f(c)) is a point of inflection of the graph of f, then either f"(c)=0 or f" is undefined at c
Critical Value
Where first derivative is 0 or undefined
Find absolute extrema
Use critical values and END POINTS in the function
Horizontal Asymptote Rules
If m>n: NO HA
If m=n: HA = co-eff of m/co-eff of n
If m<n: HA: y = 0
Mean Value Theorem
if f(x) is continuous on [a,b] and differentiable on (a,b), there is at least one point (x=c) where f'(c)= F(b)-F(a)/b-a
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.
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
Rolle's Theorem
If f(x) is continuous on the closed interval [a, b], differentiable on (a, b), and satisfies f(a) = f(b), then for some c in the interval (a, b), we have f'(c) = 0
If f (c) ≤ f (x) for every x in the domain of f, then the point (c, f (c)) is a local minimum.
FALSE...The point is an absolute minimum. Local minima must be critical numbers,
and this statement says nothing about critical numbers.
If c is a critical number of f, then f has a relative maximum, relative minimum, or an inflection point at c
FALSE...critical numbers are places where the derivative is zero or undefined. It is possible that it could be a local extrema or an inflection point, but it doesn't necessarily have to be. Keep in mind that the derivative will be undefined at any discontinuity. Therefore c could have also been a point of removable discontinuity, jump discontinuity or infinite discontinuity.
If a function is continuous and differentiable on (a,b), then f attains an absolute maximum and absolute minimum value at some values c and d in (a,b).
FALSE...the function would need to be continuous on the closed interval [a,b] in order to attain an absolute maximum and absolute minimum. Consider the function
f (x) = tan x on ⎛ − π/2 , π/2 ⎞. This is continuous on the whole interval, but has no absolute extrema as there are two vertical asymptotes. The graph tends towards −∞ as x approaches − π/2 and it tends towards ∞ as x approaches π/2 .
If f has a local maximum or minimum at c, then f '(c) = 0
FALSE...Consider the graph of f (x) = |x| . There is a local minimum at 0, but
f '(0) = DNE .
All global extrema are either local extrema or occur at the endpoints of a closed interval.
True
If f is concave up on an interval, then the graph of f ' is increasing on that interval.
True
If f (c) exists, f'(c) = 0 , and f''(c) > 0 , then there is a local maximum at x = c .
FALSE...there is a local minimum at x = c . This comes from the second derivative test.
In optimization, how do you prove that the critical value is maximum or minimum?
1. Use the number line or
2. Second derivative test
Find HA: y = (x+2)/(sqrt(x^2+3))
y=+1 and y=-1
parametric equations
A pair of equations that define the x and y coordinates of a point in terms of a third variable called a parameter.
Given x and y, how to parametricize?
Table: |t | x | y| Plug in x,y to graph
An object moving along a line through the point (x0, y0), with dx/dt = a and dy/dt = b,
has parametric equations
x = x0 + at,
y = y0 + bt.
REMEMBER: a = dx/dt, not dx/dy
b = dy/dt, not dy/dx
The instantaneous speed of a moving object is defined to be
Horizontal Asymptote for exponential functions?
y = k
(if y = e^x + k)
lim (c) = ?
x->a
c
If the function if not continuous, does the limit exist?
The limit exists if there is removable discontinuity
If a function is differentiable, is it continuous?
Yes! Differentiability => Supercontinuity
Linear Approximation Formula
L(x) = f(a) + f'(a)(x-a)
second derivative of parametrically defined curve
find first derivative, dy/dx = dy/dt / dx/dt, then find derivative of first derivative, then divide by dx/dt
OR
if f'(x)<= g'(x) for all x then f(x)<=g(x) for all x
False
This condition is true only if they start at the same point. (Racetrack Principle)
What happens if you half delta(n) while calculating integral
The difference between upper and lower estimate gets halved; more accurate prediction since velocity is measured more frequently
If f(x) is even, what is
a
∫f(x) ?
-a
a
2∫f(x)
0
If f(x) is odd, what is
a
∫f(x) ?
-a
0
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