Section 1
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∫kf(x)dx
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Date created
Mar 1, 2020
Cards (68)
Section 1
(50 cards)
∫kf(x)dx
k∫f(x)dx
d/dx[tanu]=
d/dx[tanu]= (sec²u)u'
∫csc(x)cot(x)dx
-csc(x) + C
A critical number occurs when:
f'(x)=0 or is undefined and f(x) is changing from increasing to decreasing or vice versa.
Average Rate of Change
[f(b)-f(a)]/(b-a)
d/dx [cosu]
(-sinu)(u')
∫csc²(x)dx
-cot(x) + C
∫csc(u)du
-ln|csc(u)+cot(u)| + C
∫(1/u)du
ln|u| + C
d/dx [c]
0
d/dx[e∧u]
(e^u)u'
∫tan(u)du
-ln|cos(u)| + C
lim(x→ ±∞)(ƒ(x))= If the degree of the numerator is > the degree of the denominator
Does not exist
The line y=L is a horizontal asymptote of the graph of ƒ if:
lim(x→ ±∞)(ƒ(x))= L
d/dx[ln(u)]=
u'/u
∫kdx
kx+C
d/dx[lnu]
u'/u
Quotient Rule d/dx[f(x)/g(x)]=
d/dx[f(x)/g(x)]= [g(x)f'(x)-f(x)g'(x)]/[g(x)]²
lim(x→ ±∞)(ƒ(x))= If the degree of the numerator is = the degree of the denominator
The ratio of the leading coefficients
d/dx[secu]=
d/dx[secu]= (secu·tanu)u'
d/dx[cscu]=
d/dx[cscu]= -(cscu·cotu)u'
Limit definition of a derivative
ƒ'(x)= lim(∆x→0) [f(x+∆x)-f(x)]/ (∆x)
A function is continuous if the following 3 conditions are met:
1. ƒ(c) is defined 2. lim(x→c)(ƒ(x)) exists 3. ƒ(c) = lim(x→c)(ƒ(x))
∫cot(u)du
ln|sin(u)| + C
lim(x→ ±∞)(ƒ(x))= If the degree of the numerator is < the degree of the denominator
0
lim(x→0)[(sinx)/x]=
1
∫0dx
C
lim(x→0)[(1-cosx)/x]=
0
∫sec²(x)dx
tan(x) + C
d/dx [sinu]
(cosu)(u')
When does the derivative of f(x) not exist?
f(x) is undefined, where f(x) has a sharp turn, where f(x) has a vertical tangent line
lim(x→c)(x)=
c
Product Rule: d/dx[f(x)g(x)]=
d/dx[f(x)g(x)]= f(x)g'(x)+g(x)f'(x)
If f and g are inverse functions, g'(x)=
1/ƒ'(g(x))
∫sin(x)dx
-cos(x) + C
If s(t)= Position, then s'(t) = s''(t)=
s'(t)= velocity s''(t) = acceleration
∫sec(x)tan(x)dx
sec(x) + C
d/dx[cotu]=
d/dx[cotu]= -(csc²u)u'
lim(x→c)(b)=
b
d/dx [uⁿ]
nuⁿ⁻¹u'
lim(x→c)(x^n)=
c^n
Average value of f on an interval (a,b)
1/(b-a)∫f(x)dx [NOTE: limits of integration are from a to b]
If ƒ(x) approaches ±∞ as x approaches c from the right or left, then x=c is a ________
Vertical Asymptote
Instantaneous Rate of Change at x=c of f(x)
=f'(c)
∫[f(x)±g(x)]dx
∫f(x)dx ± ∫g(x)dx
A point of inflection occurs when:
f''(x)=0 or is undefined and f(x) is changing from concave upward to concave downward or vice versa.
∫xⁿdx
[x∧(n+1)]/(n+1) + C
∫cos(x)dx
sin(x) + C
∫sec(u)du
ln|sec(u)+tan(u)| + C
When does a limit fail to exist?
1. The left handed limit does not equal the right handed limit 2. Unbounded behavior 3. Oscillating
Section 2
(18 cards)