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29 terms

Calc 1

Derivatives
STUDY
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derivative of sin(x)
= cos(x)
derivative of cos(x)
= -sin(x)
derivative of tan(x)
= sec^2(x)
derivative of sec(x)
= sec(x) tan(x)
derivative of csc(x)
= -csc(x) cot(x)
derivative of cot(x)
= -csc^2(x)
derivative of trig functions reminders
...functions that start with "c" all have negative derivatives
...tan & cot have squares
...sec & csc are themselves x tan & cot respectively
derivative of a^x
= (a^x) (ln a)

where "a" is a constant
y = ln x = log_e_x
y' = 1/(x ln e) <------ln e = 1

ex: y = log_10_4x u= 4x y=log_10_u
y' = 1/(u ln 10) x (4)
y' = 4/(4x ln 10) = 1/(x ln 10)
velocity
is the first derivative of a position function
s(t) = position function
s'(t) = velocity, v(t)
acceleration
is the second derivative of a position function
s(t) = position function
s'(t) = v(t), velocity
s''(t) = v'(t) = acceleration, a(t)
marginal cost
is the first derivative of a cost function and tell us the cost to produce one more item
rate of change of population
first derivative of a population equation/function
derivative of arccosine x
(inverse cosine)
= -1/(sqrt 1-x^2)
derivative of arcsine x
(inverse sine)
= 1/(sqrt 1-x^2)
derivative of arctangent x
(inverse tangent)
= 1/(1+x^2)
derivative of e^u
= (e^u)(u')
derivative of a^u
= (ln a)(a^u)(u')
derivative of a function
(fixed #)
f(a+h) - f(a)
f'(a) = lim h-->0 --------------------
h
derivative of a function
(number varies)
f(x+h) - f(x)
f'(x) = lim h-->0 --------------------
h
csc
1/sin
sec
1/cos
cot
cos/sin
3 requirements for a function to becontinuous at x=a
f(x) must be defined at x=a
lim x-->a f(x) must exist
f(a) = lim x-->a f(x)
point slope form
(y - y_1) = m (x - x_1)
tan
sin/cos
Parametric equations
dy/dx = y'/x'
Horizontal asymptote
y = (numerator's leading coefficient) / (denominator's leading coefficient) = 1/1 = 1
If there isn't a variable in the numerator, the ha is at 0
Vertical asymptote
set denominator = 0 and solve