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AP Calculus Bc

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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. Given velocity vectors dx/dt and dy/dt - find total distance travelled
f(x)
? v (dx/dt) + (dy/dt) over interval from a to b
critical points and endpoints
? abs[v(t)] over interval a to b

2. Converges absolutely
Alternating series converges and general term converges with another test
A function and it's derivative are in the integrand
use tangent line to approximate values of the function
? v(t) over interval a to b

3. definite integral
has limits a & b - find antiderivative - F(b) - F(a)
derivative
use tangent line to approximate values of the function
p ? R - r 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. Quotient Rule
Limit as h approaches 0 of [f(a+h)-f(a)]/h
point of inflection
velocity is negative
(uv'-vu')/v

5. Area inside one polar curve and outside another polar curve
? f(x) dx integrate over interval a to b
p ? r dx over interval a to b - where r = distance from curve to axis of revolution
Slope of secant line between two points - use to estimate instantanous rate of change at a point.
1/2 ? R - r over interval from a to b - find a & b by setting equations equal - solve for theta.

6. When f '(x) is negative - f(x) is...
f '(g(x)) g'(x)
increasing
decreasing
0/0 - 8/8 - 8*0 - 8 - 8 - 1^8 - 0 - 8

7. L'Hopitals rule
1/(b-a) ? f(x) dx on interval a to b
corner - cusp - vertical tangent - discontinuity
use to find indeterminate limits - find derivative of numerator and denominator separately then evaluate limit
y' = -1/v(1 - x)

8. Geometric series test
1/2 ? r over interval from a to b - find a & b by setting r = 0 - solve for theta
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.
? f(x) - g(x) over interval a to b - where f(x) is top function and g(x) is bottom function
general term = a1r^n - converges if -1 < r < 1

9. Integral test
if integral converges - series converges
v(dx/dt) + (dy/dt) not an integral!
y' = 1/(x lna)
Slope of tangent line at a point - value of derivative at a point

10. right riemann sum
y' = -1/v(1 - x)
decreasing
Slope of secant line between two points - use to estimate instantanous rate of change at a point.
use rectangles with right-endpoints to evaluate integrals (estimate area)

11. Alternate definition of derivative
Limit as x approaches a of [f(x)-f(a)]/(x-a)
f '(g(x)) g'(x)
logistic differential equation - M = carrying capacity
if lim as n approaches 8 of ratio of comparison series/general term is positive and finite - then series behaves like comparison series

12. slope of horizontal line
zero
? A(x) dx over interval a to b - where A(x) is the area of the given cross-section in terms of x
logistic growth equation
(uv'-vu')/v

13. P = M / (1 + Ae^(-Mkt))
logistic growth equation
Limit as x approaches a of [f(x)-f(a)]/(x-a)
logistic differential equation - M = carrying capacity
? v(t) over interval a to b

14. left riemann sum
use ratio test - set > 1 and solve absolute value equations - check endpoints
if lim as n approaches 8 of ratio of comparison series/general term is positive and finite - then series behaves like comparison series
y' = -sin(x)
use rectangles with left-endpoints to evaluate integral (estimate area)

15. 6th degree Taylor Polynomial
y' = -csc(x)cot(x)
polynomial with finite number of terms - largest exponent is 6 - find all derivatives up to the 6th derivative
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 '(g(x)) g'(x)

16. When f '(x) changes from increasing to decreasing or decreasing to increasing - f(x) has a...
point of inflection
if integral converges - series converges
y' = 1/x
increasing

17. If g(x) = ? f(t) dt on interval 2 to x - then g'(x) =...
derivative
f(x)
y' = -csc(x)cot(x)
? v (dx/dt) + (dy/dt) over interval from a to b

18. When is a function not differentiable
if terms grow without bound - series diverges
integrand is a rational function with a factorable denominator
uv' + vu'
corner - cusp - vertical tangent - discontinuity

19. Alternating series tes
1/2 ? R - r over interval from a to b - find a & b by setting equations equal - solve for theta.
critical points and endpoints
lim as n approaches zero of general term = 0 and terms decrease - series converges
logistic differential equation - M = carrying capacity

20. Formal definition of derivative
Limit as h approaches 0 of [f(a+h)-f(a)]/h
use tangent line to approximate values of the function
derivative
? v (dx/dt) + (dy/dt) over interval from a to b

21. y = tan?(x) - y' =
y' = 1/(1 + x)
lim as n approaches zero of general term = 0 and terms decrease - series converges
? v (dx/dt) + (dy/dt) over interval from a to b
lim as n approaches 8 of ratio of (n+1) term/nth term > 1 - series converges

22. y = csc(x) - y' =
y' = -csc(x)cot(x)
A function and it's derivative are in the integrand
p ? r dx over interval a to b - where r = distance from curve to axis of revolution
e^x

23. Intermediate Value Theorem
use ratio test - set > 1 and solve absolute value equations - check endpoints
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' = 1/x
concave up

24. Given v(t) find displacement
corner - cusp - vertical tangent - discontinuity
? v(t) over interval a to b
negative
y' = sec(x)tan(x)

25. Length of curve
polynomial with finite number of terms - largest exponent is 6 - find all derivatives up to the 6th derivative
? v(1 + (dy/dx)) dx over interval a to b
no limits - find antiderivative + C - use inital value to find C
A function and it's derivative are in the integrand

26. Converges conditionally
Alternating series converges and general term diverges with another test
f '(g(x)) g'(x)
f(x)
f(x) has a relative maximum

27. y = cot?(x) - y' =
Limit as h approaches 0 of [f(a+h)-f(a)]/h
y' = -csc(x)cot(x)
y' = -1/(1 + x)
use ratio test - set > 1 and solve absolute value equations - radius = center - endpoint

28. Instantenous Rate of Change
decreasing
Slope of tangent line at a point - value of derivative at a point
general term = 1/n^p - converges if p > 1
separate variables - integrate + C - use initial condition to find C - solve for y

29. [(h1 - h2)/2]*base
Alternating series converges and general term diverges with another test
Area of trapezoid
f(x) has a relative maximum
product rule

30. Find interval of convergence
use ratio test - set > 1 and solve absolute value equations - check endpoints
y' = sec(x)tan(x)
relative minimum
? f(x) dx on interval a to b = F(b) - F(a)

31. ? u dv =
uv - ? v du
1/2 ? r over interval from a to b - find a & b by setting r = 0 - solve for theta
uv' + vu'
concave up

32. average value of f(x)
general term = a1r^n - converges if -1 < r < 1
corner - cusp - vertical tangent - discontinuity
1/(b-a) ? f(x) dx on interval a to b
use ratio test - set > 1 and solve absolute value equations - check endpoints

33. Area inside polar curve
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.
derivative
1/2 ? r over interval from a to b - find a & b by setting r = 0 - solve for theta
if terms grow without bound - series diverges

34. To find particular solution to differential equation - dy/dx = x/y...
critical points and endpoints
? v(t) over interval a to b
? v (dx/dt) + (dy/dt) over interval from a to b
separate variables - integrate + C - use initial condition to find C - solve for y

35. Taylor series
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'
polynomial with infinite number of terms - includes general term
v(dx/dt) + (dy/dt) not an integral!

36. methods of integration
? f(x) dx integrate over interval a to b
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.

37. y = x cos(x) - state rule used to find derivative
use trapezoids to evaluate integrals (estimate area)
product rule
f(x)
uv' + vu'

38. y = tan(x) - y' =
Slope of tangent line at a point - value of derivative at a point
point of inflection
y' = sec(x)
use rectangles with left-endpoints to evaluate integral (estimate area)

39. Indeterminate forms
0/0 - 8/8 - 8*0 - 8 - 8 - 1^8 - 0 - 8
? f(x) dx integrate over interval a to b
y' = cos(x)
y' = sec(x)tan(x)

40. Chain Rule
? f(x) dx on interval a to b = F(b) - F(a)
f '(g(x)) g'(x)
y' = -csc(x)
quotient rule

41. y = sin?(x) - y' =
f(x) has a relative minimum
1/(b-a) ? f(x) dx on interval a to b
y' = 1/v(1 - x)
A function and it's derivative are in the integrand

42. y = ln(x) - y' =
corner - cusp - vertical tangent - discontinuity
y' = 1/x
concave up
y' = cos(x)

43. Particle is moving to the right/up
if lim as n approaches 8 of ratio of comparison series/general term is positive and finite - then series behaves like comparison series
? f(x) dx on interval a to b = F(b) - F(a)
Limit as x approaches a of [f(x)-f(a)]/(x-a)
velocity is positive

44. Product Rule
uv' + vu'
lim as n approaches 8 of ratio of (n+1) term/nth term > 1 - series converges
? A(x) dx over interval a to b - where A(x) is the area of the given cross-section in terms of x
point of inflection

45. Given velocity vectors dx/dt and dy/dt - find speed
if integral converges - series converges
? v (dx/dt) + (dy/dt) over interval from a to b
v(dx/dt) + (dy/dt) not an integral!
if terms grow without bound - series diverges

46. Volume of solid of revolution - no washer
no limits - find antiderivative + C - use inital value to find C
? v(1 + (dy/dx)) dx over interval a to b
p ? r dx over interval a to b - where r = distance from curve to axis of revolution
negative

47. slope of vertical line
use rectangles with left-endpoints to evaluate integral (estimate area)
undefined
use rectangles with right-endpoints to evaluate integrals (estimate area)
relative maximum

48. When f '(x) is decreasing - f(x) is...
general term = a1r^n - converges if -1 < r < 1
? A(x) dx over interval a to b - where A(x) is the area of the given cross-section in terms of x
concave down
f(x) has a relative minimum

49. Linearization
? v (dx/dt) + (dy/dt) over interval from a to b
y' = 1/x
lim as n approaches zero of general term = 0 and terms decrease - series converges
use tangent line to approximate values of the function

50. rate
0/0 - 8/8 - 8*0 - 8 - 8 - 1^8 - 0 - 8
y' = sec(x)
derivative
Alternating series converges and general term diverges with another test

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AP Calculus Bc

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