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. use substitution to integrate when

2. When f '(x) is negative - f(x) is...
Area of trapezoid
decreasing
chain rule
general term = a1r^n - converges if -1 < r < 1

3. y = cot?¹(x) - y' =

4. slope of vertical line
? v(t) over interval a to b
undefined
0/0 - 8/8 - 8*0 - 8 - 8 - 1^8 - 0° - 8°
Limit as x approaches a of [f(x)-f(a)]/(x-a)

5. Product Rule

6. Eatio test
lim as n approaches 8 of ratio of (n+1) term/nth term > 1 - series converges
Slope of tangent line at a point - value of derivative at a point
? v(t) over interval a to b
? f(x) dx integrate over interval a to b

7. indefinite integral
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 not an integral!
if terms grow without bound - series diverges
no limits - find antiderivative + C - use inital value to find C

8. right riemann sum
use trapezoids to evaluate integrals (estimate area)
0/0 - 8/8 - 8*0 - 8 - 8 - 1^8 - 0° - 8°
no limits - find antiderivative + C - use inital value to find C
use rectangles with right-endpoints to evaluate integrals (estimate area)

9. Intermediate Value Theorem
integrand is a rational function with a factorable denominator
relative maximum
f(x)
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.

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

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

12. L'Hopitals rule
critical points and endpoints
y' = cos(x)
use to find indeterminate limits - find derivative of numerator and denominator separately then evaluate limit
negative

13. Formal definition of derivative
Limit as h approaches 0 of [f(a+h)-f(a)]/h
a^x ln(a)
y' = sec²(x)
y' = 1/(1 + x²)

14. y = cos²(3x)
chain rule
f(x)
if terms grow without bound - series diverges
velocity is positive

15. Converges absolutely
f(x) has a relative minimum
f '(g(x)) g'(x)
Alternating series converges and general term converges with another test
y' = 1/x

16. If f '(x) = 0 and f'(x) < 0 -...
relative maximum
f(x) has a relative maximum
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.
? abs[v(t)] over interval a to b

17. P = M / (1 + Ae^(-Mkt))
y' = 1/v(1 - x²)
? f(x) - g(x) over interval a to b - where f(x) is top function and g(x) is bottom function
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
logistic growth equation

18. Linearization
use tangent line to approximate values of the function
p ? r² dx over interval a to b - where r = distance from curve to axis of revolution
critical points and endpoints
f(x)

19. Particle is moving to the right/up
? v (dx/dt)² + (dy/dt)² over interval from a to b
y' = sec²(x)
velocity is positive
relative maximum

20. methods of integration
p ? r² dx over interval a to b - where r = distance from curve to axis of revolution
substitution - parts - partial fractions
y' = -csc(x)cot(x)
use tangent line to approximate values of the function

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

22. Alternate definition of derivative
if integral converges - series converges
use ratio test - set > 1 and solve absolute value equations - check endpoints
Limit as x approaches a of [f(x)-f(a)]/(x-a)
y' = -sin(x)

23. Area between two curves
polynomial with finite number of terms - largest exponent is 6 - find all derivatives up to the 6th derivative
chain rule
? f(x) - g(x) over interval a to b - where f(x) is top function and g(x) is bottom function
separate variables - integrate + C - use initial condition to find C - solve for y

24. Volume of solid of revolution - washer
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
use ratio test - set > 1 and solve absolute value equations - check endpoints
? f(x) dx integrate over interval a to b

25. definite integral
has limits a & b - find antiderivative - F(b) - F(a)
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
substitution - parts - partial fractions

26. Given velocity vectors dx/dt and dy/dt - find total distance travelled
Slope of secant line between two points - use to estimate instantanous rate of change at a point.
use to find indeterminate limits - find derivative of numerator and denominator separately then evaluate limit
y' = -csc(x)cot(x)
? v (dx/dt)² + (dy/dt)² over interval from a to b

27. Quotient Rule

28. p-series test
general term = 1/n^p - converges if p > 1
1/(b-a) ? f(x) dx on interval a to b
chain rule
f '(g(x)) g'(x)

29. Indeterminate forms
0/0 - 8/8 - 8*0 - 8 - 8 - 1^8 - 0° - 8°
find first derivative - dy/dx = dy/dt / dx/dt - then find derivative of first derivative - then divide by dx/dt
relative maximum
A function and it's derivative are in the integrand

30. Volume of solid of revolution - no washer
polynomial with infinite number of terms - includes general term
p ? r² dx over interval a to b - where r = distance from curve to axis of revolution
? v (dx/dt)² + (dy/dt)² over interval from a to b
general term = 1/n^p - converges if p > 1

31. y = sin(x) - y' =

32. area under a curve
? f(x) dx integrate over interval a to b
? A(x) dx over interval a to b - where A(x) is the area of the given cross-section in terms of x
y' = sec²(x)
y' = -csc(x)cot(x)

33. use integration by parts when...
chain rule
two different types of functions are multiplied
if terms grow without bound - series diverges
lim as n approaches zero of general term = 0 and terms decrease - series converges

34. 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
general term = a1r^n - converges if -1 < r < 1
Limit as x approaches a of [f(x)-f(a)]/(x-a)
use ratio test - set > 1 and solve absolute value equations - check endpoints

35. 6th degree Taylor Polynomial
critical points and endpoints
polynomial with finite number of terms - largest exponent is 6 - find all derivatives up to the 6th derivative
if lim as n approaches 8 of ratio of comparison series/general term is positive and finite - then series behaves like comparison series
velocity is positive

36. Length of curve
? v(1 + (dy/dx)²) dx over interval a to b
f(x) has a relative maximum
point of inflection
1/2 ? R² - r² over interval from a to b - find a & b by setting equations equal - solve for theta.

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

38. Given v(t) find displacement
velocity is positive
? v(t) over interval a to b
substitution - parts - partial fractions
y' = -csc(x)cot(x)

39. area below x-axis is...
has limits a & b - find antiderivative - F(b) - F(a)
negative
y' = -1/(1 + x²)
logistic growth equation

40. Use partial fractions to integrate when...
velocity is negative
? abs[v(t)] over interval a to b
integrand is a rational function with a factorable denominator
1/2 ? R² - r² over interval from a to b - find a & b by setting equations equal - solve for theta.

41. Find radius of convergence
use ratio test - set > 1 and solve absolute value equations - radius = center - endpoint
has limits a & b - find antiderivative - F(b) - F(a)
use tangent line to approximate values of the function
velocity is negative

42. Area inside one polar curve and outside another polar curve
negative
concave down
? abs[v(t)] over interval a to b
1/2 ? R² - r² over interval from a to b - find a & b by setting equations equal - solve for theta.

43. Instantenous Rate of Change
general term = a1r^n - converges if -1 < r < 1
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)
relative maximum
Slope of tangent line at a point - value of derivative at a point

44. When f '(x) changes fro positive to negative - f(x) has a...
y' = sec(x)tan(x)
y' = -csc(x)cot(x)
velocity is negative
relative maximum

45. Given v(t) find total distance travelled
if terms grow without bound - series diverges
substitution - parts - partial fractions
? abs[v(t)] over interval a to b
use to find indeterminate limits - find derivative of numerator and denominator separately then evaluate limit

46. rate
no limits - find antiderivative + C - use inital value to find C
(uv'-vu')/v²
0/0 - 8/8 - 8*0 - 8 - 8 - 1^8 - 0° - 8°
derivative

47. dP/dt = kP(M - P)
point of inflection
velocity is negative
logistic differential equation - M = carrying capacity
y' = -1/(1 + x²)

48. Find interval of convergence
use ratio test - set > 1 and solve absolute value equations - check endpoints
? abs[v(t)] over interval a to b
? v(1 + (dy/dx)²) dx over interval a to b
y' = 1/x

49. Chain Rule

50. y = tan?¹(x) - y' =