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. average value of f(x)
f(x)
y' = cos(x)
(uv'-vu')/v²
1/(b-a) ? f(x) dx on interval a to b


2. Geometric series test
general term = a1r^n - converges if -1 < r < 1
has limits a & b - find antiderivative - F(b) - F(a)
Area of trapezoid
relative maximum


3. L'Hopitals rule
use to find indeterminate limits - find derivative of numerator and denominator separately then evaluate limit
y' = -csc²(x)
? f(x) - g(x) over interval a to b - where f(x) is top function and g(x) is bottom function
lim as n approaches zero of general term = 0 and terms decrease - series converges


4. If g(x) = ? f(t) dt on interval 2 to x - then g'(x) =...
f(x) has a relative maximum
f(x)
Limit as h approaches 0 of [f(a+h)-f(a)]/h
y' = -csc²(x)


5. use substitution to integrate when

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6. y = ln(x)/x² - state rule used to find derivative
f(x)
y' = 1/(x lna)
y' = -csc²(x)
quotient rule


7. y = cos(x) - y' =

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8. y = cos²(3x)
chain rule
decreasing
find first derivative - dy/dx = dy/dt / dx/dt - then find derivative of first derivative - then divide by dx/dt
use rectangles with right-endpoints to evaluate integrals (estimate area)


9. y = sec(x) - y' =

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10. slope of horizontal line
product rule
zero
A function and it's derivative are in the integrand
y' = 1/(1 + x²)


11. mean value theorem

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

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13. Product Rule

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14. slope of vertical line
find first derivative - dy/dx = dy/dt / dx/dt - then find derivative of first derivative - then divide by dx/dt
undefined
Limit as h approaches 0 of [f(a+h)-f(a)]/h
two different types of functions are multiplied


15. To draw a slope field - plug (x -y) coordinates into differential equation...
undefined
general term = 1/n^p - converges if p > 1
y' = 1/(x lna)
draw short segments representing slope at each point


16. Limit comparison test
if lim as n approaches 8 of ratio of comparison series/general term is positive and finite - then series behaves like comparison series
point of inflection
polynomial with infinite number of terms - includes general term
y' = 1/(x lna)


17. When f '(x) is increasing - f(x) is...
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.
y' = sec(x)tan(x)
concave up
derivative


18. Area between two curves
? f(x) - g(x) over interval a to b - where f(x) is top function and g(x) is bottom function
decreasing
Limit as x approaches a of [f(x)-f(a)]/(x-a)
y' = -csc²(x)


19. y = tan?¹(x) - y' =

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20. To find particular solution to differential equation - dy/dx = x/y...
if lim as n approaches 8 of ratio of comparison series/general term is positive and finite - then series behaves like comparison series
(uv'-vu')/v²
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)


21. Given velocity vectors dx/dt and dy/dt - find speed
velocity is negative
v(dx/dt)² + (dy/dt)² not an integral!
use ratio test - set > 1 and solve absolute value equations - check endpoints
Alternating series converges and general term diverges with another test


22. When f '(x) is positive - f(x) is...
use tangent line to approximate values of the function
use ratio test - set > 1 and solve absolute value equations - radius = center - endpoint
relative maximum
increasing


23. To find absolute maximum on closed interval [a - b] - you must consider...
two different types of functions are multiplied
y' = -1/v(1 - x²)
y' = cos(x)
critical points and endpoints


24. Integral test
? abs[v(t)] over interval a to b
if integral converges - series converges
logistic growth equation
? v(1 + (dy/dx)²) dx over interval a to b


25. Volume of solid of revolution - washer
has limits a & b - find antiderivative - F(b) - F(a)
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
if terms grow without bound - series diverges
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.


26. Given velocity vectors dx/dt and dy/dt - find total distance travelled
critical points and endpoints
f(x)
? v (dx/dt)² + (dy/dt)² over interval from a to b
y' = cos(x)


27. Particle is moving to the right/up
velocity is positive
negative
? v (dx/dt)² + (dy/dt)² over interval from a to b
Alternating series converges and general term converges with another test


28. Second derivative of parametrically defined curve
e^x
y' = cos(x)
find first derivative - dy/dx = dy/dt / dx/dt - then find derivative of first derivative - then divide by dx/dt
two different types of functions are multiplied


29. y = sin?¹(x) - y' =

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30. Chain Rule

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31. Fundamental Theorem of Calculus
substitution - parts - partial fractions
? f(x) dx on interval a to b = F(b) - F(a)
Slope of secant line between two points - use to estimate instantanous rate of change at a point.
y' = -sin(x)


32. methods of integration
? abs[v(t)] over interval a to b
use rectangles with left-endpoints to evaluate integral (estimate area)
substitution - parts - partial fractions
critical points and endpoints


33. 6th degree Taylor Polynomial
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)
polynomial with finite number of terms - largest exponent is 6 - find all derivatives up to the 6th derivative
y' = -sin(x)
logistic differential equation - M = carrying capacity


34. left riemann sum
point of inflection
quotient rule
v(dx/dt)² + (dy/dt)² not an integral!
use rectangles with left-endpoints to evaluate integral (estimate area)


35. Find radius of convergence
relative maximum
use ratio test - set > 1 and solve absolute value equations - radius = center - endpoint
integrand is a rational function with a factorable denominator
critical points and endpoints


36. Formal definition of derivative
? v(1 + (dy/dx)²) dx over interval a to b
product rule
Limit as h approaches 0 of [f(a+h)-f(a)]/h
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.


37. If f '(x) = 0 and f'(x) < 0 -...
f(x)
f(x) has a relative maximum
use trapezoids to evaluate integrals (estimate area)
? A(x) dx over interval a to b - where A(x) is the area of the given cross-section in terms of x


38. area below x-axis is...
negative
quotient rule
lim as n approaches zero of general term = 0 and terms decrease - series converges
1/2 ? r² over interval from a to b - find a & b by setting r = 0 - solve for theta


39. When is a function not differentiable
corner - cusp - vertical tangent - discontinuity
use rectangles with left-endpoints to evaluate integral (estimate area)
speed
relative maximum


40. dP/dt = kP(M - P)
? v(1 + (dy/dx)²) dx over interval a to b
? v (dx/dt)² + (dy/dt)² over interval from a to b
lim as n approaches 8 of ratio of (n+1) term/nth term > 1 - series converges
logistic differential equation - M = carrying capacity


41. Taylor series
relative maximum
polynomial with infinite number of terms - includes general term
1/2 ? R² - r² over interval from a to b - find a & b by setting equations equal - solve for theta.
negative


42. Given v(t) find total distance travelled
v(dx/dt)² + (dy/dt)² not an integral!
? abs[v(t)] over interval a to b
use ratio test - set > 1 and solve absolute value equations - radius = center - endpoint
logistic differential equation - M = carrying capacity


43. Alternate definition of derivative
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)
Limit as x approaches a of [f(x)-f(a)]/(x-a)
chain rule


44. y = csc(x) - y' =

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45. [(h1 - h2)/2]*base
Area of trapezoid
1/2 ? r² over interval from a to b - find a & b by setting r = 0 - solve for theta
relative minimum
y' = 1/v(1 - x²)


46. ? u dv =
uv - ? v du
1/2 ? r² over interval from a to b - find a & b by setting r = 0 - solve for theta
concave up
Area of trapezoid


47. absolute value of velocity
relative maximum
1/2 ? R² - r² over interval from a to b - find a & b by setting equations equal - solve for theta.
speed
y' = sec(x)tan(x)


48. Average Rate of Change
Slope of secant line between two points - use to estimate instantanous rate of change at a point.
find first derivative - dy/dx = dy/dt / dx/dt - then find derivative of first derivative - then divide by dx/dt
critical points and endpoints
polynomial with infinite number of terms - includes general term


49. trapezoidal rule
use trapezoids to evaluate integrals (estimate area)
? v(1 + (dy/dx)²) dx over interval a to b
if terms grow without bound - series diverges
has limits a & b - find antiderivative - F(b) - F(a)


50. Area inside polar curve
chain rule
A function and it's derivative are in the integrand
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.
1/2 ? r² over interval from a to b - find a & b by setting r = 0 - solve for theta