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. Area inside polar curve
if integral converges - series converges
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
1/2 ? r² over interval from a to b - find a & b by setting r = 0 - solve for theta
chain rule


2. Alternating series tes
use tangent line to approximate values of the function
lim as n approaches zero of general term = 0 and terms decrease - series converges
decreasing
1/2 ? r² over interval from a to b - find a & b by setting r = 0 - solve for theta


3. Particle is moving to the right/up
velocity is positive
general term = 1/n^p - converges if p > 1
has limits a & b - find antiderivative - F(b) - F(a)
1/2 ? R² - r² over interval from a to b - find a & b by setting equations equal - solve for theta.


4. Fundamental Theorem of Calculus
positive
Alternating series converges and general term converges with another test
v(dx/dt)² + (dy/dt)² not an integral!
? f(x) dx on interval a to b = F(b) - F(a)


5. rate
relative minimum
y' = cos(x)
negative
derivative


6. Area between two curves
negative
? f(x) - g(x) over interval a to b - where f(x) is top function and g(x) is bottom function
0/0 - 8/8 - 8*0 - 8 - 8 - 1^8 - 0° - 8°
Alternating series converges and general term diverges with another test


7. Quotient Rule


8. absolute value of velocity
(uv'-vu')/v²
zero
speed
integrand is a rational function with a factorable denominator


9. [(h1 - h2)/2]*base
Limit as h approaches 0 of [f(a+h)-f(a)]/h
Area of trapezoid
Alternating series converges and general term converges with another test
1/2 ? R² - r² over interval from a to b - find a & b by setting equations equal - solve for theta.


10. y = cot?¹(x) - y' =


11. To draw a slope field - plug (x -y) coordinates into differential equation...
v(dx/dt)² + (dy/dt)² not an integral!
speed
draw short segments representing slope at each point
y' = -sin(x)


12. Converges absolutely
substitution - parts - partial fractions
zero
Alternating series converges and general term converges with another test
? v(1 + (dy/dx)²) dx over interval a to b


13. Indeterminate forms
0/0 - 8/8 - 8*0 - 8 - 8 - 1^8 - 0° - 8°
if lim as n approaches 8 of ratio of comparison series/general term is positive and finite - then series behaves like comparison series
y' = -1/v(1 - x²)
? v (dx/dt)² + (dy/dt)² over interval from a to b


14. Formal definition of derivative
use rectangles with right-endpoints to evaluate integrals (estimate area)
? f(x) dx integrate over interval a to b
general term = a1r^n - converges if -1 < r < 1
Limit as h approaches 0 of [f(a+h)-f(a)]/h


15. y = tan(x) - y' =


16. Alternate definition of derivative
decreasing
f(x) has a relative maximum
y' = 1/v(1 - x²)
Limit as x approaches a of [f(x)-f(a)]/(x-a)


17. right riemann sum
speed
use ratio test - set > 1 and solve absolute value equations - check endpoints
use rectangles with right-endpoints to evaluate integrals (estimate area)
concave down


18. When f '(x) changes from negative to positive - f(x) has a...
relative minimum
lim as n approaches 8 of ratio of (n+1) term/nth term > 1 - series converges
logistic differential equation - M = carrying capacity
critical points and endpoints


19. When f '(x) changes fro positive to negative - f(x) has a...
Slope of tangent line at a point - value of derivative at a point
relative maximum
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)
? f(x) dx on interval a to b = F(b) - F(a)


20. To find absolute maximum on closed interval [a - b] - you must consider...
? v (dx/dt)² + (dy/dt)² over interval from a to b
critical points and endpoints
zero
? abs[v(t)] over interval a to b


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


22. Use partial fractions to integrate when...
relative minimum
(uv'-vu')/v²
logistic differential equation - M = carrying capacity
integrand is a rational function with a factorable denominator


23. When f '(x) is negative - f(x) is...
decreasing
relative minimum
y' = 1/(x lna)
speed


24. dP/dt = kP(M - P)
logistic differential equation - M = carrying capacity
use ratio test - set > 1 and solve absolute value equations - check endpoints
f(x)
product rule


25. Area inside one polar curve and outside another polar curve
undefined
y' = -csc²(x)
1/2 ? R² - r² over interval from a to b - find a & b by setting equations equal - solve for theta.
y' = sec(x)tan(x)


26. Geometric series test
general term = a1r^n - converges if -1 < r < 1
f(x) has a relative minimum
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)
substitution - parts - partial fractions


27. To find particular solution to differential equation - dy/dx = x/y...
integrand is a rational function with a factorable denominator
separate variables - integrate + C - use initial condition to find C - solve for y
logistic growth equation
critical points and endpoints


28. Given velocity vectors dx/dt and dy/dt - find total distance travelled
chain rule
y' = 1/(x lna)
? v (dx/dt)² + (dy/dt)² over interval from a to b
Area of trapezoid


29. y = x cos(x) - state rule used to find derivative
product rule
general term = 1/n^p - converges if p > 1
y' = 1/(1 + x²)
y' = -sin(x)


30. 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
negative
? f(x) dx integrate over interval a to b
polynomial with finite number of terms - largest exponent is 6 - find all derivatives up to the 6th derivative


31. Chain Rule


32. nth term test
lim as n approaches 8 of ratio of (n+1) term/nth term > 1 - series converges
if terms grow without bound - series diverges
if integral converges - series converges
f(x)


33. When f '(x) is positive - f(x) is...
increasing
y' = 1/(x lna)
speed
find first derivative - dy/dx = dy/dt / dx/dt - then find derivative of first derivative - then divide by dx/dt


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


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


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


37. When f '(x) is increasing - f(x) is...
concave up
velocity is positive
y' = sec(x)tan(x)
f(x) has a relative minimum


38. y = sin?¹(x) - y' =


39. Volume of solid of revolution - no washer
? v(t) over interval a to b
p ? r² dx over interval a to b - where r = distance from curve to axis of revolution
negative
general term = a1r^n - converges if -1 < r < 1


40. definite integral
has limits a & b - find antiderivative - F(b) - F(a)
a^x ln(a)
? A(x) dx over interval a to b - where A(x) is the area of the given cross-section in terms of x
? f(x) - g(x) over interval a to b - where f(x) is top function and g(x) is bottom function


41. When f '(x) is decreasing - f(x) is...
y' = -sin(x)
A function and it's derivative are in the integrand
velocity is negative
concave down


42. y = log (base a) x - y' =


43. Volume of solid of revolution - washer
e^x
zero
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
concave down


44. use integration by parts when...
use rectangles with right-endpoints to evaluate integrals (estimate area)
two different types of functions are multiplied
relative minimum
use ratio test - set > 1 and solve absolute value equations - check endpoints


45. Given velocity vectors dx/dt and dy/dt - find speed
undefined
v(dx/dt)² + (dy/dt)² not an integral!
no limits - find antiderivative + C - use inital value to find C
lim as n approaches 8 of ratio of (n+1) term/nth term > 1 - series converges


46. Average Rate of Change
critical points and endpoints
find first derivative - dy/dx = dy/dt / dx/dt - then find derivative of first derivative - then divide by dx/dt
velocity is negative
Slope of secant line between two points - use to estimate instantanous rate of change at a point.


47. Eatio test
? A(x) dx over interval a to b - where A(x) is the area of the given cross-section in terms of x
decreasing
two different types of functions are multiplied
lim as n approaches 8 of ratio of (n+1) term/nth term > 1 - series converges


48. y = ln(x)/x² - state rule used to find derivative
quotient rule
uv - ? v du
? v (dx/dt)² + (dy/dt)² over interval from a to b
f(x) has a relative minimum


49. If f '(x) = 0 and f'(x) > 0 -...
general term = 1/n^p - converges if p > 1
? f(x) dx on interval a to b = F(b) - F(a)
f(x) has a relative minimum
? f(x) - g(x) over interval a to b - where f(x) is top function and g(x) is bottom function


50. y = cos²(3x)
relative maximum
y' = -csc²(x)
chain rule
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