Calc 1 Flashcards
A function f(x) has a removable discontinuity at x = a if
- f is not defined/not continuous at x = a
- f(a) could be defined/redefined so that new function is continuous at x = a
a function f(x) is said to have a jump discontinuity at x = a if
- the limit from both sides exists
- the left and right side limits are not equal
Studying Continuity, find:
Domain
Largest Intervals (right & left continous, continous)
Discontinuity
(limits, classify, remove if possible)
need limit symbol until…
you plug in 0 for h
Power Rule
Derivative of f(x) = x^n IS nx^(n-1) for any real number n
Constant Rule
Derivative of f(x) times constant = Derivative of (c * f(x))
Sum and Difference Rules
If f(x) and g(x) are both differentiatiable, then:
Derivative of [f(x) +/- g(x)] = Derivative of f(x) +/- Derivative of g(x)
Derivative of a constant is
0
normal line
perpendicular to tangent line
lim as x approaches 0 of (cosx - 1)/x
is 0
lim as x appr. 0 of sinx/x
= 1
derivative of the sin(x)
= cos(x)
derivative of cos(x)
= -sin(x)
typically as you take derivatives
functions get simpler…
velocity is first derivative of position, s
acceleration derivative of velocity
momentum of a moving object is
the product of the objects mass and velocity
Newton
Kg * (m/s^2)
Product Rule
If f and g are both differentiatable, then the derivative of f(x)g(x) = f(x)derivative of g(x) + g(x)*derivative of f(x)
Quotient Rule
If f and g are both differentiable, then the derivative of [f(x)/g(x)] = [g(x)derivativef(x) - f(x)derivativeg(x)] / g(x)^2
Derivative of numerator has to come before derivative of denominator
Denominator is squared on bottom
derivative of 1/g(x)
- derivativeg(x)/g^2(x)
Derivative of ln(x)
1/x
derivative of a^x
ln(a) * a^x
derivative of e^x
e^x !!!
derivative of log(small a)(of x)
= 1 / (ln(a)*x)
y’ =
dy/dx
f’(x) =
1 / g’(f(x))
d/dx[sin^-1x] = 1/(sqrt(1-x^2))
d/dx[cos^-1(x)] =
-1/[sqrt(1-x^2)]
d/dx[tan^-1(x)] =
1/[1 + x^2]
d/dx[tan^-1(x)] =
1/[1 + x^2]
ln(a^b) = b * ln(a)
y = b^x, 0<b<1
limb^x as x approaches inf. = 0
limb^x as x approaches -inf. = inf.
y = b^x, b > 1
limb^x as x approaches inf. = inf.
limb^x as x approaches - inf. = 0
f(x) = a^x is multiplicative over any interval of length k with factor a^k:
f(y + k) = a^k * f(y)
e^ln6 =
6
/inf. = 0??
Log of a constant seems to be 0
Also, ln(10x) = ln(10) + ln(x)
s(u) = ln f(u)
s’(u) = f’(u) / f(u)
f’(u) = f(u) * der. of ln f(u)
ln (x/y) = lnx - lny
ln(xy) = lnx + lny
limit of ratio = ratio of limits if
Limits are finite and one on bottom is not 0
Top limit infinity and bottom > 0 OR top limit -infinity and bottom < 0 (+inf)
If signs dont match… -inf.
If top limit is finite and bottom limit is +/- inf., ghen ratio of limits is 0
These are all determinate forms.
What if (see previous)
L and M = 0
L = +/- inf. and M = +/- inf.
L is not 0 and M = 0
0/0 and inf/inf – see L’Hopital’s Rule
L/0:, ratio of limits does not exist unless bottom function does not equal 0 for all x near a… leads to essential discontinuity
If bottom function greater than 0 for all x near a, limit is pos or neg inf. (Depends on sign of L)
When g is always neg. for all x near a, limit is also pos or neg inf. (Sign opposite L)
(THINK!)
IF SIGN X CHANGES NEAR A, THERE IS NO LIMIT
All three cases: Limit may or may not exist
Indeterminate powers
0^0; 1^inf. ; inf.^0
differentiablilty implies continuity
concave upward
smile… slope increasing
f’(x) increasing
f’‘(x) > 0
concave downward
frown… slope decreasing
f’(x) decreasing
f’‘(x) < 0
derivative of anti derivative
= the function!!
To find general antiderivqtive, just
Add C to your antiderivative
Summation notation for Riemann sums…
If the funct. is non-negative and continous in the interval, the Area under funct. Graph and above x axis is A = limas n approaches infinity (rest of summation notation) (27:30)
This the definite integral
distance = velocity * (change in time)
think about the units and plotting on a graph and how you can use riemann sums!!!
Definite integral =
Limit of Reimann sum
