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Test your basic knowledge |

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. ? u dv =
increasing
? f(x) dx integrate over interval a to b
v(dx/dt)² + (dy/dt)² not an integral!
uv - ? v du

2. area above x-axis is...
y' = 1/(x lna)
? A(x) dx over interval a to b - where A(x) is the area of the given cross-section in terms of x
positive
derivative

3. Given v(t) find total distance travelled
? abs[v(t)] over interval a to b
uv' + vu'
positive
0/0 - 8/8 - 8*0 - 8 - 8 - 1^8 - 0° - 8°

4. Converges conditionally
Alternating series converges and general term diverges with another test
1/2 ? r² over interval from a to b - find a & b by setting r = 0 - solve for theta
concave up
? abs[v(t)] over interval a to b

5. area under a curve
1/(b-a) ? f(x) dx on interval a to b
? f(x) dx integrate over interval a to b
Limit as h approaches 0 of [f(a+h)-f(a)]/h
Limit as x approaches a of [f(x)-f(a)]/(x-a)

6. y = cos²(3x)
uv' + vu'
chain rule
Limit as h approaches 0 of [f(a+h)-f(a)]/h
general term = a1r^n - converges if -1 < r < 1

7. Use partial fractions to integrate when...
Alternating series converges and general term diverges with another test
general term = a1r^n - converges if -1 < r < 1
speed
integrand is a rational function with a factorable denominator

8. nth term test
use ratio test - set > 1 and solve absolute value equations - check endpoints
derivative
if terms grow without bound - series diverges
Slope of tangent line at a point - value of derivative at a point

9. y = ln(x)/x² - state rule used to find derivative
Slope of tangent line at a point - value of derivative at a point
uv - ? v du
? abs[v(t)] over interval a to b
quotient rule

10. Volume of solid of revolution - no washer
y' = 1/v(1 - x²)
substitution - parts - partial fractions
p ? r² dx over interval a to b - where r = distance from curve to axis of revolution
if integral converges - series converges

11. Linearization
draw short segments representing slope at each point
increasing
use tangent line to approximate values of the function
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)

12. P = M / (1 + Ae^(-Mkt))
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.
(uv'-vu')/v²
logistic growth equation
derivative

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

14. Area inside polar curve
Area of trapezoid
product rule
1/2 ? r² over interval from a to b - find a & b by setting r = 0 - solve for theta
integrand is a rational function with a factorable denominator

15. rate
undefined
derivative
logistic differential equation - M = carrying capacity
(uv'-vu')/v²

16. When f '(x) is positive - f(x) is...
increasing
chain rule
if lim as n approaches 8 of ratio of comparison series/general term is positive and finite - then series behaves like comparison series
polynomial with finite number of terms - largest exponent is 6 - find all derivatives up to the 6th derivative

17. When f '(x) is decreasing - f(x) is...
1/2 ? R² - r² over interval from a to b - find a & b by setting equations equal - solve for theta.
zero
point of inflection
concave down

18. Formal definition of derivative
y' = -1/(1 + x²)
lim as n approaches 8 of ratio of (n+1) term/nth term > 1 - series converges
Limit as h approaches 0 of [f(a+h)-f(a)]/h
derivative

19. When f '(x) is negative - f(x) is...
use ratio test - set > 1 and solve absolute value equations - check endpoints
f '(g(x)) g'(x)
Slope of tangent line at a point - value of derivative at a point
decreasing

20. When f '(x) is increasing - f(x) is...
use rectangles with left-endpoints to evaluate integral (estimate area)
A function and it's derivative are in the integrand
y' = -sin(x)
concave up

21. trapezoidal rule
a^x ln(a)
use trapezoids to evaluate integrals (estimate area)
relative minimum
increasing

22. Particle is moving to the left/down
lim as n approaches 8 of ratio of (n+1) term/nth term > 1 - series converges
velocity is negative
Slope of secant line between two points - use to estimate instantanous rate of change at a point.
Area of trapezoid

23. Find interval of convergence
chain rule
Area of trapezoid
Limit as x approaches a of [f(x)-f(a)]/(x-a)
use ratio test - set > 1 and solve absolute value equations - check endpoints

24. Taylor series
polynomial with infinite number of terms - includes general term
y' = -sin(x)
? f(x) - g(x) over interval a to b - where f(x) is top function and g(x) is bottom function
draw short segments representing slope at each point

25. dP/dt = kP(M - P)
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)
logistic differential equation - M = carrying capacity
relative maximum
1/2 ? r² over interval from a to b - find a & b by setting r = 0 - solve for theta

26. Length of parametric curve
y' = sec²(x)
speed
Area of trapezoid
? v (dx/dt)² + (dy/dt)² over interval from a to b

27. Average Rate of Change
Slope of secant line between two points - use to estimate instantanous rate of change at a point.
draw short segments representing slope at each point
use rectangles with right-endpoints to evaluate integrals (estimate area)
? f(x) dx integrate over interval a to b

28. When f '(x) changes from increasing to decreasing or decreasing to increasing - f(x) has a...
polynomial with finite number of terms - largest exponent is 6 - find all derivatives up to the 6th derivative
if integral converges - series converges
point of inflection
Alternating series converges and general term converges with another test

29. Volume of solid with base in the plane and given cross-section
undefined
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
? 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) has a relative maximum

30. Integral test
use to find indeterminate limits - find derivative of numerator and denominator separately then evaluate limit
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
? v(1 + (dy/dx)²) dx over interval a to b

31. Given velocity vectors dx/dt and dy/dt - find total distance travelled
? v (dx/dt)² + (dy/dt)² over interval from a to b
undefined
polynomial with finite number of terms - largest exponent is 6 - find all derivatives up to the 6th derivative
integrand is a rational function with a factorable denominator

32. To find particular solution to differential equation - dy/dx = x/y...
separate variables - integrate + C - use initial condition to find C - solve for y
two different types of functions are multiplied
substitution - parts - partial fractions
if integral converges - series converges

33. use integration by parts when...
relative minimum
y' = 1/(1 + x²)
two different types of functions are multiplied
Area of trapezoid

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

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

36. If f '(x) = 0 and f'(x) > 0 -...
y' = -sin(x)
velocity is positive
no limits - find antiderivative + C - use inital value to find C
f(x) has a relative minimum

37. Chain Rule

38. y = tan?¹(x) - y' =

39. y = cot?¹(x) - y' =

40. left riemann sum
use rectangles with left-endpoints to evaluate integral (estimate area)
find first derivative - dy/dx = dy/dt / dx/dt - then find derivative of first derivative - then divide by dx/dt
? 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

41. y = sin?¹(x) - y' =

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

43. absolute value of velocity
Limit as x approaches a of [f(x)-f(a)]/(x-a)
speed
general term = 1/n^p - converges if p > 1
velocity is negative

44. Indeterminate forms
0/0 - 8/8 - 8*0 - 8 - 8 - 1^8 - 0° - 8°
derivative
y' = 1/(x lna)
a^x ln(a)

45. right riemann sum
Alternating series converges and general term diverges with another test
use rectangles with right-endpoints to evaluate integrals (estimate area)
point of inflection
corner - cusp - vertical tangent - discontinuity

46. Area between two curves
p ? r² dx over interval a to b - where r = distance from curve to axis of revolution
? f(x) - g(x) over interval a to b - where f(x) is top function and g(x) is bottom function
y' = -1/(1 + x²)
general term = a1r^n - converges if -1 < r < 1

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

48. To find absolute maximum on closed interval [a - b] - you must consider...
y' = -csc(x)cot(x)
Alternating series converges and general term diverges with another test
A function and it's derivative are in the integrand
critical points and endpoints

49. slope of horizontal line
relative maximum
(uv'-vu')/v²
zero
use ratio test - set > 1 and solve absolute value equations - radius = center - endpoint

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