AP Calculus BC Exam_1 Flashcards

Terms Definitions
Quotient Rule
area below x-axis is
y = cos²(3x)
chain rule
slope of vertical line
slope of horizontal line
Product Rule
uv' + vu'
[(h1 - h2)/2]*base
area of trapezoid
Midpoint Rule
f(a+ΔX(k+1/2)) = f((Xk+Xk+1)/2)
Total Area: Mn=ΔX(f((X0+X1)/2)+...f((Xn-1+Xn)/2) = ΔXsum(f((Xk+Xk+1)/2),n-1,k=0)
methods of integration
substitution, parts, partial fractions
converges absolutely
alternating series converges and general term converges with another test
Integral of: (csc(x))^2 dx
-cot(x) + c
x = ? (Polar)
x = r*cos(@)
integral test
if integral converges, series converges
When f '(x) is positive, f(x) is
y = sec(x), y' =
y' = sec(x)tan(x)
y = tan(x), y' =
y' = sec²(x)
y = e^x, y' =
y' = e^x
Squeeze Theorem:
Integral (1/x) = divergent, Integral (1/x^2) = convergent. If less than the former it is convergent, if more than the latter it is divergent.
y = csc(x), y' =
y' = -csc(x)cot(x)
trapezoidal rule
use trapezoids to evaluate integrals (estimate area)
y = cos(x), y' =
y' = -sin(x)
arc length
pi INT(a to b) SQRT(1 +(dy/dx)^2)dx
Riemann Sums
ΔXsum(f(Xk),n-1,k=0), where Xk is any sample ppoint in the kth subinterval
right riemann sum
use rectangles with right-endpoints to evaluate integrals (estimate area)
Particle is moving to the right/up
velocity is positive
First Derivative
Shows the rate of change over time
Integral of: csc(x) dx
ln|csc(x) - cot(x)| + c
left riemann sum
use rectangles with left-endpoints to evaluate integral (estimate area)
When f '(x) is increasing, f(x) is
concave up
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)
To draw a slope field,
plug (x,y) coordinates into differential equation, draw short segments representing slope at each point
y = tan⁻¹(x), y' =
y' = 1/(1 + x²)
6th degree Taylor Polynomial
polynomial with finite number of terms, largest exponent is 6, find all derivatives up to the 6th derivative
When f '(x) changes from negative to positive, f(x) has a
relative minimum
y = sin⁻¹(x), y' =
y' = 1/√(1 - x²)
l'Hospital's Rule
lim (f(x)) = lim (f'(x)) = lim (f''(x)) ...
indeterminate forms
0/0, ∞/∞, ∞*0, ∞ - ∞, 1^∞, 0⁰, ∞⁰
y = cot⁻¹(x), y' =
y' = -1/(1 + x²)
Which functions are continuous everywhere?
Polynomial, Rational, functions made up of continuous functions
i.e. f + g , f - g , f * g, f/g ( if g dne 0)
Implicit Differentiation
1. Separate y and x terms
2. Differentiate each term of the equation with respect to x
3. Factor out (dy/dx) on the left side of the equation
4. Solve for (dy/dx)
dP/dt = kP(M - P)
logistic differential equation, M = carrying capacity
Formal definition of derivative
limit as h approaches 0 of [f(a+h)-f(a)]/h
Average Rate of Change
Slope of secant line between two points, use to estimate instantanous rate of change at a point.
alternating series test
lim as n approaches zero of general term = 0 and terms decrease, series converges
length of parametric curve
∫ √ (dx/dt)² + (dy/dt)² over interval from a to b
ratio test
lim as n approaches ∞ of ratio of (n+1) term/nth term > 1, series converges
Instantenous Rate of Change
Slope of tangent line at a point, value of derivative at a point
use integration by parts when
two different types of functions are multiplied
Theorem: If the series sigma a(n) from 1 - infinite is convergent...
lim a(n) = 0
given velocity vectors dx/dt and dy/dt, find speed
√(dx/dt)² + (dy/dt)² not an integral!
Derivatives of Exponential and Logarithmic Functions
Let u be a differentiable function.
(e^u)' = e^u
(a^u)' = a^u lnu (du/dx)
use partial fractions to integrate when
integrand is a rational function with a factorable denominator
find interval of convergence
use ratio test, set > 1 and solve absolute value equations, check endpoints
second derivative of parametrically defined curve
find first derivative, dy/dx = dy/dt / dx/dt, then find derivative of first derivative, then divide by dx/dt
Sum & Difference Rules for derivatives
You can add or subtract derivatives
(g +/- f)' = g' +/- f'
Second Derivative Test for Relative Extrema
If f'(c)=0 and f''(c)<0, then f(c) is a relative maximum
If f'(c)=0 and f''(c)> 0, then f(c) is a relative minimum
If f'(c)=0 and f''(c)=0, then f(c) is a relative maximum
average value of f(x)
= 1/(b-a) ∫ f(x) dx on interval a to b
area inside polar curve
1/2 ∫ r² over interval from a to b, find a & b by setting r = 0, solve for theta
For geometric series: sigma (a*r^(n-1)) from 1 to infinite...
Convergent if |r| < 1 and sum is a/(1-r)
Divergent if |r| >= 1
If g(x) = ∫ f(t) dt on interval 2 to x, then g'(x) =
g'(x) = f(x)
Test for Increasing/Decreasing Functions
If f'(x) &lt; 0 on [a,b] , then f(x) is decreasing on [a,b]
If f'(x) &gt; 0 on [a,b] , then f(x) is increasing on [a,b]
If f'(x) = 0 on [a,b] , then f(x) is a constant on [a,b]
Fundamental Theorem of Calculus
∫ f(x) dx on interval a to b = F(b) - F(a)
First Derivative Test for Relative Extrema
If f'(x) changes from pos to neg at x=c, then f has a relative maximum at c
If f'(x) changes from neg to pos at x=c, then f has a relative minimum at c
given rate equation, R(t) and inital condition when
t = a, R(t) = y₁ find final value when t = b
y₁ + Δy = y
Δy = ∫ R(t) over interval a to b
volume of solid of revolution - washer
π ∫ R² - r² dx over interval a to b, where R = distance from outside curve to axis of revolution, r = distance from inside curve to axis of revolution
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