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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. y = ln(x)/x - state rule used to find derivative
quotient rule
v(dx/dt) + (dy/dt) not an integral!
y' = cos(x)
polynomial with finite number of terms - largest exponent is 6 - find all derivatives up to the 6th derivative

2. Area inside polar curve
Slope of tangent line at a point - value of derivative at a point
logistic differential equation - M = carrying capacity
1/2 ? r over interval from a to b - find a & b by setting r = 0 - solve for theta
has limits a & b - find antiderivative - F(b) - F(a)

3. Geometric series test
0/0 - 8/8 - 8*0 - 8 - 8 - 1^8 - 0 - 8
general term = a1r^n - converges if -1 < r < 1
y' = 1/v(1 - x)
if integral converges - series converges

4. Eatio test
find first derivative - dy/dx = dy/dt / dx/dt - then find derivative of first derivative - then divide by dx/dt
zero
lim as n approaches 8 of ratio of (n+1) term/nth term > 1 - series converges
0/0 - 8/8 - 8*0 - 8 - 8 - 1^8 - 0 - 8

5. To find particular solution to differential equation - dy/dx = x/y...
f(x)
velocity is positive
separate variables - integrate + C - use initial condition to find C - solve for y
uv' + vu'

6. [(h1 - h2)/2]*base
? f(x) dx on interval a to b = F(b) - F(a)
y' = cos(x)
? abs[v(t)] over interval a to b
Area of trapezoid

7. Given v(t) find displacement
use tangent line to approximate values of the function
? v(t) over interval a to b
? f(x) - g(x) over interval a to b - where f(x) is top function and g(x) is bottom function
logistic growth equation

8. Converges conditionally
Alternating series converges and general term diverges with another test
concave up
? v(1 + (dy/dx)) dx over interval a to b
y' = sec(x)

9. Fundamental Theorem of Calculus
e^x
? f(x) dx on interval a to b = F(b) - F(a)
polynomial with infinite number of terms - includes general term
use rectangles with right-endpoints to evaluate integrals (estimate area)

10. y = cos?(x) - y' =
quotient rule
general term = a1r^n - converges if -1 < r < 1
? f(x) dx on interval a to b = F(b) - F(a)
y' = -1/v(1 - x)

11. use substitution to integrate when
if lim as n approaches 8 of ratio of comparison series/general term is positive and finite - then series behaves like comparison series
A function and it's derivative are in the integrand
y' = -1/(1 + x)
chain rule

12. Area between two curves
uv - ? v du
? f(x) - g(x) over interval a to b - where f(x) is top function and g(x) is bottom 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)
y' = -1/v(1 - x)

13. Converges absolutely
Alternating series converges and general term converges with another test
logistic growth equation
use rectangles with left-endpoints to evaluate integral (estimate area)
use rectangles with right-endpoints to evaluate integrals (estimate area)

14. area above x-axis is...
quotient rule
substitution - parts - partial fractions
A function and it's derivative are in the integrand
positive

15. y = cot?(x) - y' =
y' = -csc(x)
y' = -1/(1 + x)
a^x ln(a)
lim as n approaches 8 of ratio of (n+1) term/nth term > 1 - series converges

16. Length of parametric curve
integrand is a rational function with a factorable denominator
y' = -sin(x)
? v (dx/dt) + (dy/dt) over interval from a to b
y' = -1/(1 + x)

17. If f '(x) = 0 and f'(x) < 0 -...
f(x) has a relative maximum
use tangent line to approximate values of the function
if terms grow without bound - series diverges
a^x ln(a)

18. If f '(x) = 0 and f'(x) > 0 -...
velocity is positive
Slope of secant line between two points - use to estimate instantanous rate of change at a point.
concave down
f(x) has a relative minimum

19. area under a curve
(uv'-vu')/v
y' = 1/(1 + x)
0/0 - 8/8 - 8*0 - 8 - 8 - 1^8 - 0 - 8
? f(x) dx integrate over interval a to b

20. y = cot(x) - y' =
f '(g(x)) g'(x)
speed
y' = -csc(x)
logistic differential equation - M = carrying capacity

21. area below x-axis is...
? f(x) dx integrate over interval a to b
? v (dx/dt) + (dy/dt) over interval from a to b
negative
zero

22. y = cos(x) - y' =
y' = -1/(1 + x)
y' = -sin(x)
product rule
? v(t) over interval a to b

23. Given velocity vectors dx/dt and dy/dt - find speed
Limit as h approaches 0 of [f(a+h)-f(a)]/h
v(dx/dt) + (dy/dt) not an integral!
y' = 1/(x lna)
zero

24. trapezoidal rule
y' = -sin(x)
use trapezoids to evaluate integrals (estimate area)
use to find indeterminate limits - find derivative of numerator and denominator separately then evaluate limit
use ratio test - set > 1 and solve absolute value equations - check endpoints

25. y = tan(x) - y' =
f(x) has a relative maximum
chain rule
y' = sec(x)
negative

26. Indeterminate forms
? v(t) over interval a to b
? f(x) dx on interval a to b = F(b) - F(a)
speed
0/0 - 8/8 - 8*0 - 8 - 8 - 1^8 - 0 - 8

27. Length of curve
no limits - find antiderivative + C - use inital value to find C
? v(1 + (dy/dx)) dx over interval a to b
use to find indeterminate limits - find derivative of numerator and denominator separately then evaluate limit
f(x) has a relative minimum

28. Volume of solid of revolution - washer
use ratio test - set > 1 and solve absolute value equations - radius = center - endpoint
Alternating series converges and general term converges with another test
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
Limit as x approaches a of [f(x)-f(a)]/(x-a)

29. Instantenous Rate of Change
has limits a & b - find antiderivative - F(b) - F(a)
two different types of functions are multiplied
negative
Slope of tangent line at a point - value of derivative at a point

30. Integral test
derivative
if integral converges - series converges
separate variables - integrate + C - use initial condition to find C - solve for y
uv' + vu'

31. rate
logistic differential equation - M = carrying capacity
y' = 1/(1 + x)
derivative
lim as n approaches 8 of ratio of (n+1) term/nth term > 1 - series converges

32. indefinite integral
negative
Limit as h approaches 0 of [f(a+h)-f(a)]/h
y' = 1/(1 + x)
no limits - find antiderivative + C - use inital value to find C

33. To find absolute maximum on closed interval [a - b] - you must consider...
Limit as x approaches a of [f(x)-f(a)]/(x-a)
Area of trapezoid
critical points and endpoints
polynomial with infinite number of terms - includes general term

34. absolute value of velocity
p ? r dx over interval a to b - where r = distance from curve to axis of revolution
general term = a1r^n - converges if -1 < r < 1
speed
uv' + vu'

35. Linearization
polynomial with finite number of terms - largest exponent is 6 - find all derivatives up to the 6th derivative
f(x) has a relative minimum
use tangent line to approximate values of the function
f(x) has a relative maximum

36. Use partial fractions to integrate when...
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/(x lna)
integrand is a rational function with a factorable denominator
y' = -csc(x)cot(x)

37. y = ln(x) - y' =
velocity is positive
y' = cos(x)
y' = 1/x
logistic differential equation - M = carrying capacity

38. use integration by parts when...
? f(x) dx on interval a to b = F(b) - F(a)
? abs[v(t)] over interval a to b
find first derivative - dy/dx = dy/dt / dx/dt - then find derivative of first derivative - then divide by dx/dt
two different types of functions are multiplied

39. When f '(x) is negative - f(x) is...
use ratio test - set > 1 and solve absolute value equations - check endpoints
decreasing
Slope of secant line between two points - use to estimate instantanous rate of change at a point.
relative minimum

40. y = sec(x) - y' =
y' = sec(x)tan(x)
chain rule
y' = -sin(x)
? abs[v(t)] over interval a to b

41. When f '(x) is positive - f(x) is...
? v (dx/dt) + (dy/dt) over interval from a to b
separate variables - integrate + C - use initial condition to find C - solve for y
increasing
v(dx/dt) + (dy/dt) not an integral!

42. definite integral
Limit as h approaches 0 of [f(a+h)-f(a)]/h
Area of trapezoid
? f(x) dx on interval a to b = F(b) - F(a)
has limits a & b - find antiderivative - F(b) - F(a)

43. 6th degree Taylor Polynomial
y' = -csc(x)
use rectangles with left-endpoints to evaluate integral (estimate area)
polynomial with finite number of terms - largest exponent is 6 - find all derivatives up to the 6th derivative
y' = 1/(1 + x)

44. left riemann sum
use rectangles with left-endpoints to evaluate integral (estimate area)
two different types of functions are multiplied
y' = 1/v(1 - x)
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)

45. slope of vertical line
f(x)
? f(x) dx integrate over interval a to b
? v (dx/dt) + (dy/dt) over interval from a to b
undefined

46. Alternating series tes
Slope of tangent line at a point - value of derivative at a point
y' = sec(x)tan(x)
y' = -sin(x)
lim as n approaches zero of general term = 0 and terms decrease - series converges

47. Intermediate Value Theorem
relative minimum
lim as n approaches 8 of ratio of (n+1) term/nth term > 1 - series converges
Slope of secant line between two points - use to estimate instantanous rate of change at a point.
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.

48. nth term test
velocity is negative
quotient rule
if terms grow without bound - series diverges
? f(x) - g(x) over interval a to b - where f(x) is top function and g(x) is bottom function

49. Average Rate of Change
Slope of secant line between two points - use to estimate instantanous rate of change at a point.
if terms grow without bound - series diverges
critical points and endpoints
velocity is negative

50. Find radius of convergence
use ratio test - set > 1 and solve absolute value equations - radius = center - endpoint
polynomial with infinite number of terms - includes general term
? v(1 + (dy/dx)) dx over interval a to b
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.

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AP Calculus Bc

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