AP Calculus Bc

Instructions:

• Answer 50 questions in 15 minutes.
• If you are not ready to take this test, you can study here.
• Match each statement with the correct term.
• Don't refresh. All questions and answers are randomly picked and ordered every time you load a test.

This is a study tool. The 3 wrong answers for each question are randomly chosen from answers to other questions. So, you might find at times the answers obvious, but you will see it re-enforces your understanding as you take the test each time.

1. nth term test
use tangent line to approximate values of the function
1/(b-a) ? f(x) dx on interval a to b
Alternating series converges and general term converges with another test
if terms grow without bound - series diverges


2. trapezoidal rule
1/2 ? r² over interval from a to b - find a & b by setting r = 0 - solve for theta
use trapezoids to evaluate integrals (estimate area)
? v (dx/dt)² + (dy/dt)² over interval from a to b
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.


3. [(h1 - h2)/2]*base
concave down
Area of trapezoid
f '(g(x)) g'(x)
draw short segments representing slope at each point


4. y = cos²(3x)
no limits - find antiderivative + C - use inital value to find C
Alternating series converges and general term converges with another test
chain rule
lim as n approaches zero of general term = 0 and terms decrease - series converges


5. area under a curve
? f(x) dx integrate over interval a to b
velocity is negative
chain rule
separate variables - integrate + C - use initial condition to find C - solve for y


6. average value of f(x)
1/(b-a) ? f(x) dx on interval a to b
critical points and endpoints
use to find indeterminate limits - find derivative of numerator and denominator separately then evaluate limit
? v (dx/dt)² + (dy/dt)² over interval from a to b


7. ? u dv =
y' = -1/(1 + x²)
uv - ? v du
Limit as x approaches a of [f(x)-f(a)]/(x-a)
speed


8. 6th degree Taylor Polynomial
f(x)
polynomial with finite number of terms - largest exponent is 6 - find all derivatives up to the 6th derivative
? f(x) dx on interval a to b = F(b) - F(a)
y' = -1/(1 + x²)


9. methods of integration
0/0 - 8/8 - 8*0 - 8 - 8 - 1^8 - 0° - 8°
y' = 1/(x lna)
p ? r² dx over interval a to b - where r = distance from curve to axis of revolution
substitution - parts - partial fractions


10. mean value theorem

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11. y = e^x - y' = y' =
use rectangles with right-endpoints to evaluate integrals (estimate area)
y' = -sin(x)
e^x
logistic growth equation


12. p-series test
general term = 1/n^p - converges if p > 1
use trapezoids to evaluate integrals (estimate area)
no limits - find antiderivative + C - use inital value to find C
integrand is a rational function with a factorable denominator


13. Area between two curves
? f(x) dx integrate over interval a to b
no limits - find antiderivative + C - use inital value to find C
? f(x) - g(x) over interval a to b - where f(x) is top function and g(x) is bottom function
f(x)


14. definite integral
f(x) has a relative maximum
has limits a & b - find antiderivative - F(b) - F(a)
corner - cusp - vertical tangent - discontinuity
f(x)


15. y = tan?¹(x) - y' =

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16. Volume of solid with base in the plane and given cross-section
separate variables - integrate + C - use initial condition to find C - solve for y
y' = sec²(x)
? A(x) dx over interval a to b - where A(x) is the area of the given cross-section in terms of x
chain rule


17. When f '(x) is positive - f(x) is...
? abs[v(t)] over interval a to b
product rule
no limits - find antiderivative + C - use inital value to find C
increasing


18. absolute value of velocity
? v (dx/dt)² + (dy/dt)² over interval from a to b
speed
Alternating series converges and general term converges with another test
concave up


19. Particle is moving to the right/up
velocity is positive
if terms grow without bound - series diverges
undefined
y' = -csc²(x)


20. Limit comparison test
? abs[v(t)] over interval a to b
if lim as n approaches 8 of ratio of comparison series/general term is positive and finite - then series behaves like comparison series
? v (dx/dt)² + (dy/dt)² over interval from a to b
critical points and endpoints


21. Average Rate of Change
? f(x) - g(x) over interval a to b - where f(x) is top function and g(x) is bottom function
use ratio test - set > 1 and solve absolute value equations - check endpoints
Limit as h approaches 0 of [f(a+h)-f(a)]/h
Slope of secant line between two points - use to estimate instantanous rate of change at a point.


22. When f '(x) is increasing - f(x) is...
concave up
relative maximum
substitution - parts - partial fractions
p ? 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


23. When f '(x) changes fro positive to negative - f(x) has a...
lim as n approaches zero of general term = 0 and terms decrease - series converges
? v(1 + (dy/dx)²) dx over interval a to b
relative maximum
y' = 1/(1 + x²)


24. Integral test
product rule
? A(x) dx over interval a to b - where A(x) is the area of the given cross-section in terms of x
if integral converges - series converges
point of inflection


25. Converges conditionally
if terms grow without bound - series diverges
uv - ? v du
Alternating series converges and general term diverges with another test
y' = sec(x)tan(x)


26. left riemann sum
product rule
lim as n approaches 8 of ratio of (n+1) term/nth term > 1 - series converges
use ratio test - set > 1 and solve absolute value equations - radius = center - endpoint
use rectangles with left-endpoints to evaluate integral (estimate area)


27. Volume of solid of revolution - no washer
Alternating series converges and general term converges with another test
p ? r² dx over interval a to b - where r = distance from curve to axis of revolution
draw short segments representing slope at each point
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)


28. Linearization
use tangent line to approximate values of the function
critical points and endpoints
general term = 1/n^p - converges if p > 1
general term = a1r^n - converges if -1 < r < 1


29. Product Rule

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30. When f '(x) changes from negative to positive - f(x) has a...
critical points and endpoints
Alternating series converges and general term converges with another test
if terms grow without bound - series diverges
relative minimum


31. Length of parametric curve
f(x) has a relative maximum
find first derivative - dy/dx = dy/dt / dx/dt - then find derivative of first derivative - then divide by dx/dt
? v (dx/dt)² + (dy/dt)² over interval from a to b
relative maximum


32. Find radius of convergence
? A(x) dx over interval a to b - where A(x) is the area of the given cross-section in terms of x
substitution - parts - partial fractions
lim as n approaches 8 of ratio of (n+1) term/nth term > 1 - series converges
use ratio test - set > 1 and solve absolute value equations - radius = center - endpoint


33. Quotient Rule

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34. y = cot(x) - y' =

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35. slope of vertical line
undefined
? v(1 + (dy/dx)²) dx over interval a to b
concave down
Limit as h approaches 0 of [f(a+h)-f(a)]/h


36. To find absolute maximum on closed interval [a - b] - you must consider...
use tangent line to approximate values of the function
critical points and endpoints
find first derivative - dy/dx = dy/dt / dx/dt - then find derivative of first derivative - then divide by dx/dt
1/2 ? R² - r² over interval from a to b - find a & b by setting equations equal - solve for theta.


37. Geometric series test
negative
polynomial with finite number of terms - largest exponent is 6 - find all derivatives up to the 6th derivative
general term = a1r^n - converges if -1 < r < 1
product rule


38. right riemann sum
derivative
a^x ln(a)
use rectangles with right-endpoints to evaluate integrals (estimate area)
polynomial with infinite number of terms - includes general term


39. Area inside one polar curve and outside another polar curve
? v (dx/dt)² + (dy/dt)² over interval from a to b
undefined
1/2 ? R² - r² over interval from a to b - find a & b by setting equations equal - solve for theta.
decreasing


40. Formal definition of derivative
increasing
Limit as h approaches 0 of [f(a+h)-f(a)]/h
p ? r² dx over interval a to b - where r = distance from curve to axis of revolution
substitution - parts - partial fractions


41. When f '(x) changes from increasing to decreasing or decreasing to increasing - f(x) has a...
corner - cusp - vertical tangent - discontinuity
polynomial with infinite number of terms - includes general term
point of inflection
Alternating series converges and general term diverges with another test


42. dP/dt = kP(M - P)
y' = 1/v(1 - x²)
? v (dx/dt)² + (dy/dt)² over interval from a to b
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.
logistic differential equation - M = carrying capacity


43. Instantenous Rate of Change
Slope of tangent line at a point - value of derivative at a point
two different types of functions are multiplied
Limit as x approaches a of [f(x)-f(a)]/(x-a)
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.


44. When f '(x) is decreasing - f(x) is...
? v (dx/dt)² + (dy/dt)² over interval from a to b
lim as n approaches zero of general term = 0 and terms decrease - series converges
concave down
y' = -sin(x)


45. use substitution to integrate when

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46. If f '(x) = 0 and f'(x) > 0 -...
concave down
separate variables - integrate + C - use initial condition to find C - solve for y
zero
f(x) has a relative minimum


47. To find particular solution to differential equation - dy/dx = x/y...
separate variables - integrate + C - use initial condition to find C - solve for y
Limit as x approaches a of [f(x)-f(a)]/(x-a)
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)
increasing


48. L'Hopitals rule
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)
use to find indeterminate limits - find derivative of numerator and denominator separately then evaluate limit
y' = -csc²(x)
y' = 1/(1 + x²)


49. indefinite integral
Slope of secant line between two points - use to estimate instantanous rate of change at a point.
relative maximum
? v (dx/dt)² + (dy/dt)² over interval from a to b
no limits - find antiderivative + C - use inital value to find C


50. Given v(t) find total distance travelled
no limits - find antiderivative + C - use inital value to find C
? A(x) dx over interval a to b - where A(x) is the area of the given cross-section in terms of x
A function and it's derivative are in the integrand
? abs[v(t)] over interval a to b