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AP Calculus Bc

Instructions:

Answer 50 questions in 15 minutes.
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Match each statement with the correct term.
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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. y = tan?¹(x) - y' =

2. If f '(x) = 0 and f'(x) > 0 -...
negative
f(x) has a relative minimum
y' = 1/(1 + x²)
Alternating series converges and general term diverges with another test

3. 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' = -1/v(1 - x²)
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

4. If g(x) = ? f(t) dt on interval 2 to x - then g'(x) =...
f(x)
concave down
y' = -sin(x)
undefined

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

6. When f '(x) changes from negative to positive - f(x) has a...
relative minimum
0/0 - 8/8 - 8*0 - 8 - 8 - 1^8 - 0° - 8°
concave down
use trapezoids to evaluate integrals (estimate area)

7. Second derivative of parametrically defined curve
use ratio test - set > 1 and solve absolute value equations - check endpoints
y' = -1/v(1 - x²)
? f(x) dx on interval a to b = F(b) - F(a)
find first derivative - dy/dx = dy/dt / dx/dt - then find derivative of first derivative - then divide by dx/dt

8. To find particular solution to differential equation - dy/dx = x/y...
speed
uv' + vu'
uv - ? v du
separate variables - integrate + C - use initial condition to find C - solve for y

9. Alternating series tes
e^x
use trapezoids to evaluate integrals (estimate area)
lim as n approaches zero of general term = 0 and terms decrease - series converges
0/0 - 8/8 - 8*0 - 8 - 8 - 1^8 - 0° - 8°

10. rate
relative minimum
f '(g(x)) g'(x)
logistic growth equation
derivative

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

12. p-series test
? v(1 + (dy/dx)²) dx over interval a to b
quotient rule
use to find indeterminate limits - find derivative of numerator and denominator separately then evaluate limit
general term = 1/n^p - converges if p > 1

13. Formal definition of derivative
general term = a1r^n - converges if -1 < r < 1
Limit as h approaches 0 of [f(a+h)-f(a)]/h
logistic differential equation - M = carrying capacity
logistic growth equation

14. Limit comparison test
Alternating series converges and general term converges with another test
positive
if lim as n approaches 8 of ratio of comparison series/general term is positive and finite - then series behaves like comparison series
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

15. average value of f(x)
use rectangles with left-endpoints to evaluate integral (estimate area)
1/(b-a) ? f(x) dx on interval a to b
use trapezoids to evaluate integrals (estimate area)
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

16. Area inside one polar curve and outside another polar curve
1/2 ? r² over interval from a to b - find a & b by setting r = 0 - solve for theta
concave down
1/2 ? R² - r² over interval from a to b - find a & b by setting equations equal - solve for theta.
zero

17. definite integral
substitution - parts - partial fractions
1/2 ? R² - r² over interval from a to b - find a & b by setting equations equal - solve for theta.
has limits a & b - find antiderivative - F(b) - F(a)
? f(x) dx integrate over interval a to b

18. mean value theorem

19. Chain Rule

20. y = x cos(x) - state rule used to find derivative
find first derivative - dy/dx = dy/dt / dx/dt - then find derivative of first derivative - then divide by dx/dt
y' = sec²(x)
f '(g(x)) g'(x)
product rule

21. Intermediate Value Theorem
e^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.
y' = -sin(x)
Alternating series converges and general term diverges with another test

22. right riemann sum
y' = cos(x)
use rectangles with right-endpoints to evaluate integrals (estimate area)
? v (dx/dt)² + (dy/dt)² over interval from a to b
general term = 1/n^p - converges if p > 1

23. Given v(t) find total distance travelled
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(dx/dt)² + (dy/dt)² not an integral!
? abs[v(t)] over interval a to b
f(x)

24. Product Rule

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

26. dP/dt = kP(M - P)
integrand is a rational function with a factorable denominator
Alternating series converges and general term converges with another test
y' = -sin(x)
logistic differential equation - M = carrying capacity

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

28. To find absolute maximum on closed interval [a - b] - you must consider...
critical points and endpoints
point of inflection
use rectangles with right-endpoints to evaluate integrals (estimate area)
Slope of tangent line at a point - value of derivative at a point

29. 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
Limit as h approaches 0 of [f(a+h)-f(a)]/h
f(x)
product rule

30. Given v(t) find displacement
point of inflection
negative
? v(t) over interval a to b
Area of trapezoid

31. When f '(x) is negative - f(x) is...
concave down
a^x ln(a)
decreasing
zero

32. Converges absolutely
concave up
zero
Alternating series converges and general term converges with another test
use tangent line to approximate values of the function

33. nth term test
Slope of secant line between two points - use to estimate instantanous rate of change at a point.
? f(x) dx integrate over interval a to b
if terms grow without bound - series diverges
y' = -sin(x)

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

35. When f '(x) changes fro positive to negative - f(x) has a...
a^x ln(a)
y' = -csc(x)cot(x)
relative maximum
increasing

36. L'Hopitals rule
lim as n approaches zero of general term = 0 and terms decrease - series converges
uv - ? v du
(uv'-vu')/v²
use to find indeterminate limits - find derivative of numerator and denominator separately then evaluate limit

37. Eatio test
y' = cos(x)
use rectangles with right-endpoints to evaluate integrals (estimate area)
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

38. Integral test
a^x ln(a)
y' = -1/(1 + x²)
y' = cos(x)
if integral converges - series converges

39. [(h1 - h2)/2]*base
? v (dx/dt)² + (dy/dt)² over interval from a to b
1/2 ? r² over interval from a to b - find a & b by setting r = 0 - solve for theta
Slope of secant line between two points - use to estimate instantanous rate of change at a point.
Area of trapezoid

40. Find interval of convergence
use ratio test - set > 1 and solve absolute value equations - check endpoints
? f(x) dx on interval a to b = F(b) - F(a)
f(x) has a relative maximum
Alternating series converges and general term diverges with another test

41. P = M / (1 + Ae^(-Mkt))
two different types of functions are multiplied
logistic growth equation
point of inflection
quotient rule

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

43. Indeterminate forms
y' = 1/x
0/0 - 8/8 - 8*0 - 8 - 8 - 1^8 - 0° - 8°
decreasing
if terms grow without bound - series diverges

44. y = ln(x)/x² - state rule used to find derivative
has limits a & b - find antiderivative - F(b) - F(a)
Slope of secant line between two points - use to estimate instantanous rate of change at a point.
quotient rule
undefined

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

46. Converges conditionally
Alternating series converges and general term diverges with another test
product rule
a^x ln(a)
(uv'-vu')/v²

47. Geometric series test
general term = a1r^n - converges if -1 < r < 1
Slope of secant line between two points - use to estimate instantanous rate of change at a point.
? v (dx/dt)² + (dy/dt)² over interval from a to b
critical points and endpoints

48. left riemann sum
velocity is negative
y' = -sin(x)
use rectangles with left-endpoints to evaluate integral (estimate area)
y' = sec(x)tan(x)

49. y = cos?¹(x) - y' =

50. area above x-axis is...
1/2 ? R² - r² over interval from a to b - find a & b by setting equations equal - solve for theta.
concave up
Slope of secant line between two points - use to estimate instantanous rate of change at a point.
positive