Section 1
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y = cos(x), y' =
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Date created
Mar 1, 2020
Cards (67)
Section 1
(50 cards)
y = cos(x), y' =
y' = -sin(x)
Particle is moving to the right/up
velocity is positive
mean value theorem
if f(x) is continuous and differentiable, slope of tangent line equals slope of secant line at least once in the interval (a, b) f '(c) = [f(b) - f(a)]/(b - a)
y = log (base a) x, y' =
y' = 1/(x lna)
rate
derivative
y = e^x, y' =
y' = e^x
When f '(x) is increasing, f(x) is
concave up
If g(x) = ∫ f(t) dt on interval 2 to x, then g'(x) =
g'(x) = f(x)
y = tan⁻¹(x), y' =
y' = 1/(1 + x²)
y = x cos(x), state rule used to find derivative
product rule
Intermediate Value Theorem
If f(1)=-4 and f(6)=9, then there must be a x-value between 1 and 6 where f crosses the x-axis.
Alternate definition of derivative
limit as x approaches a of [f(x)-f(a)]/(x-a)
Fundamental Theorem of Calculus
∫ f(x) dx on interval a to b = F(b) - F(a)
right riemann sum
use rectangles with right-endpoints to evaluate integrals (estimate area)
left riemann sum
use rectangles with left-endpoints to evaluate integral (estimate area)
y = a^x, y' =
y' = a^x ln(a)
y = sin⁻¹(x), y' =
y' = 1/√(1 - x²)
When f '(x) changes from increasing to decreasing or decreasing to increasing, f(x) has a
point of inflection
y = csc(x), y' =
y' = -csc(x)cot(x)
When f '(x) is positive, f(x) is
increasing
To find absolute maximum on closed interval [a, b], you must consider...
critical points and endpoints
trapezoidal rule
use trapezoids to evaluate integrals (estimate area)
y = cot⁻¹(x), y' =
y' = -1/(1 + x²)
If f '(x) = 0 and f"(x) > 0,
f(x) has a relative minimum
Product Rule
uv' + vu'
If f '(x) = 0 and f"(x) < 0,
f(x) has a relative maximum
area under a curve
∫ f(x) dx integrate over interval a to b
average value of f(x)
= 1/(b-a) ∫ f(x) dx on interval a to b
y = sec(x), y' =
y' = sec(x)tan(x)
y = cos²(3x)
chain rule
[(h1 + h2)/2]*base
area of trapezoid
y = sin(x), y' =
y' = cos(x)
y = ln(x), y' =
y' = 1/x
y = tan(x), y' =
y' = sec²(x)
y = cos⁻¹(x), y' =
y' = -1/√(1 - x²)
Formal definition of derivative
limit as h approaches 0 of [f(a+h)-f(a)]/h
y = cot(x), y' =
y' = -csc²(x)
When f '(x) is decreasing, f(x) is
concave down
When is a function not differentiable
corner, cusp, vertical tangent, discontinuity
Average Rate of Change
Slope of secant line between two points, use to estimate instantanous rate of change at a point.
Instantenous Rate of Change
Slope of tangent line at a point, value of derivative at a point
When f '(x) changes fro positive to negative, f(x) has a
relative maximum
Linearization
use tangent line to approximate values of the function
When f '(x) is negative, f(x) is
decreasing
Particle is moving to the left/down
velocity is negative
y = ln(x)/x², state rule used to find derivative
quotient rule
When f '(x) changes from negative to positive, f(x) has a
relative minimum
absolute value of velocity
speed
Quotient Rule
(uv'-vu')/v²
Chain Rule
f '(g(x)) g'(x)
Section 2
(17 cards)