BC Flashcards
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
Formal definition of derivative
limit as h approaches 0 of [f(a+h)-f(a)]/h
Alternate definition of derivative
limit as x approaches a of [f(x)-f(a)]/(x-a)
When f ‘(x) is positive, f(x) is
increasing
When f ‘(x) is negative, f(x) is
decreasing
When f ‘(x) changes from negative to positive, f(x) has a
relative minimum
When f ‘(x) changes fro positive to negative, f(x) has a
relative maximum
When f ‘(x) is increasing, f(x) is
concave up
When f ‘(x) is decreasing, f(x) is
concave down
When f ‘(x) changes from increasing to decreasing or decreasing to increasing, f(x) has a
point of inflection
When is a function not differentiable
corner, cusp, vertical tangent, discontinuity
Product Rule
uv’ + vu’
Quotient Rule
(uv’-vu’)/v²
Chain Rule
f ‘(g(x)) g’(x)
y = x cos(x), state rule used to find derivative
product rule
y = ln(x)/x², state rule used to find derivative
quotient rule
absolute value of velocity
speed
y = sin(x), y’ =
y’ = cos(x)
y = cos(x), y’ =
y’ = -sin(x)
y = tan(x), y’ =
y’ = sec²(x)
y = csc(x), y’ =
y’ = -csc(x)cot(x)
y = sec(x), y’ =
y’ = sec(x)tan(x)
y = cot(x), y’ =
y’ = -csc²(x)
y = sin⁻¹(x), y’ =
y’ = 1/√(1 - x²)
y = cos⁻¹(x), y’ =
y’ = -1/√(1 - x²)
y = tan⁻¹(x), y’ =
y’ = 1/(1 + x²)
y = cot⁻¹(x), y’ =
y’ = -1/(1 + x²)
y = e^x, y’ =
y’ = e^x
y = a^x, y’ =
y’ = a^x ln(a)
y = ln(x), y’ =
y’ = 1/x
y = log (base a) x, y’ =
y’ = 1/(x lna)
To find absolute maximum on closed interval [a, b], you must consider…
critical points and endpoints
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)
If f ‘(x) = 0 and f”(x) > 0,
f(x) has a relative minimum
If f ‘(x) = 0 and f”(x)
f(x) has a relative maximum
Linearization
use tangent line to approximate values of the function
rate
derivative
left riemann sum
use rectangles with left-endpoints to evaluate integral (estimate area)
right riemann sum
use rectangles with right-endpoints to evaluate integrals (estimate area)
trapezoidal rule
use trapezoids to evaluate integrals (estimate area)
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.
