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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. Length of parametric curve
use ratio test - set > 1 and solve absolute value equations - check endpoints
logistic growth equation
? v (dx/dt)² + (dy/dt)² over interval from a to b
Limit as h approaches 0 of [f(a+h)-f(a)]/h

2. If f '(x) = 0 and f'(x) < 0 -...
f(x) has a relative maximum
decreasing
1/2 ? R² - r² over interval from a to b - find a & b by setting equations equal - solve for theta.
? v(1 + (dy/dx)²) dx over interval a to b

3. dP/dt = kP(M - P)
logistic differential equation - M = carrying capacity
A function and it's derivative are in the integrand
Limit as h approaches 0 of [f(a+h)-f(a)]/h
? abs[v(t)] over interval a to b

4. definite integral
concave down
velocity is positive
use rectangles with right-endpoints to evaluate integrals (estimate area)
has limits a & b - find antiderivative - F(b) - F(a)

5. Volume of solid of revolution - washer
(uv'-vu')/v²
y' = -csc²(x)
1/(b-a) ? f(x) dx on interval a to b
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

6. 6th degree Taylor Polynomial
use ratio test - set > 1 and solve absolute value equations - check endpoints
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
negative

7. Converges absolutely
uv - ? v du
negative
Alternating series converges and general term converges with another test
f(x) has a relative minimum

8. Quotient Rule

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

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

11. left riemann sum
Slope of tangent line at a point - value of derivative at a point
concave up
use rectangles with left-endpoints to evaluate integral (estimate area)
logistic growth equation

12. Geometric series test
if lim as n approaches 8 of ratio of comparison series/general term is positive and finite - then series behaves like comparison series
Area of trapezoid
? v (dx/dt)² + (dy/dt)² over interval from a to b
general term = a1r^n - converges if -1 < r < 1

13. If g(x) = ? f(t) dt on interval 2 to x - then g'(x) =...
use ratio test - set > 1 and solve absolute value equations - check endpoints
f(x)
y' = 1/x
speed

14. Fundamental Theorem of Calculus
e^x
? f(x) dx on interval a to b = F(b) - F(a)
y' = sec(x)tan(x)
polynomial with finite number of terms - largest exponent is 6 - find all derivatives up to the 6th derivative

15. To draw a slope field - plug (x -y) coordinates into differential equation...
draw short segments representing slope at each point
logistic differential equation - M = carrying capacity
y' = -csc(x)cot(x)
relative minimum

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

17. average value of f(x)
1/(b-a) ? f(x) dx on interval a to b
y' = -sin(x)
draw short segments representing slope at each point
? abs[v(t)] over interval a to b

18. When f '(x) is decreasing - f(x) is...
logistic differential equation - M = carrying capacity
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)
concave down
? v(1 + (dy/dx)²) dx over interval a to b

19. trapezoidal rule
general term = a1r^n - converges if -1 < r < 1
use trapezoids to evaluate integrals (estimate area)
two different types of functions are multiplied
undefined

20. Indeterminate forms
y' = -sin(x)
y' = cos(x)
0/0 - 8/8 - 8*0 - 8 - 8 - 1^8 - 0° - 8°
? A(x) dx over interval a to b - where A(x) is the area of the given cross-section in terms of x

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

22. p-series test
y' = -csc²(x)
general term = 1/n^p - converges if p > 1
f(x) has a relative minimum
positive

23. rate
polynomial with infinite number of terms - includes general term
f(x)
derivative
decreasing

24. Alternating series tes
if terms grow without bound - series diverges
use to find indeterminate limits - find derivative of numerator and denominator separately then evaluate limit
lim as n approaches zero of general term = 0 and terms decrease - series converges
relative minimum

25. Alternate definition of derivative
Limit as x approaches a of [f(x)-f(a)]/(x-a)
Alternating series converges and general term diverges with another test
? v (dx/dt)² + (dy/dt)² over interval from a to b
? f(x) - g(x) over interval a to b - where f(x) is top function and g(x) is bottom function

26. When is a function not differentiable
0/0 - 8/8 - 8*0 - 8 - 8 - 1^8 - 0° - 8°
e^x
relative maximum
corner - cusp - vertical tangent - discontinuity

27. To find absolute maximum on closed interval [a - b] - you must consider...
general term = a1r^n - converges if -1 < r < 1
Alternating series converges and general term diverges with another test
critical points and endpoints
e^x

28. To find particular solution to differential equation - dy/dx = x/y...
f(x) has a relative maximum
y' = cos(x)
separate variables - integrate + C - use initial condition to find C - solve for y
if integral converges - series converges

29. Converges conditionally
use ratio test - set > 1 and solve absolute value equations - radius = center - endpoint
? abs[v(t)] over interval a to b
Alternating series converges and general term diverges with another test
polynomial with infinite number of terms - includes general term

30. y = x cos(x) - state rule used to find derivative
concave up
v(dx/dt)² + (dy/dt)² not an integral!
y' = -1/(1 + x²)
product rule

31. Particle is moving to the right/up
velocity is positive
negative
Alternating series converges and general term diverges with another test
e^x

32. y = a^x - y' = y' =
a^x ln(a)
A function and it's derivative are in the integrand
1/2 ? R² - r² over interval from a to b - find a & b by setting equations equal - solve for theta.
y' = 1/(x lna)

33. y = cos?¹(x) - y' =

34. Average Rate of Change
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.
1/2 ? r² over interval from a to b - find a & b by setting r = 0 - solve for theta
undefined

35. Length of curve
? v(1 + (dy/dx)²) dx over interval a to b
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
Limit as x approaches a of [f(x)-f(a)]/(x-a)
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.

36. When f '(x) changes fro positive to negative - f(x) has a...
Area of trapezoid
if lim as n approaches 8 of ratio of comparison series/general term is positive and finite - then series behaves like comparison series
y' = sec(x)tan(x)
relative maximum

37. y = tan?¹(x) - y' =

38. P = M / (1 + Ae^(-Mkt))
y' = -1/(1 + x²)
find first derivative - dy/dx = dy/dt / dx/dt - then find derivative of first derivative - then divide by dx/dt
zero
logistic growth equation

39. Intermediate Value Theorem
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' = sec(x)tan(x)
? v (dx/dt)² + (dy/dt)² over interval from a to b
lim as n approaches 8 of ratio of (n+1) term/nth term > 1 - series converges

40. use integration by parts when...
two different types of functions are multiplied
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.
if lim as n approaches 8 of ratio of comparison series/general term is positive and finite - then series behaves like comparison series
if terms grow without bound - series diverges

41. slope of vertical line
A function and it's derivative are in the integrand
general term = a1r^n - converges if -1 < r < 1
undefined
critical points and endpoints

42. When f '(x) changes from negative to positive - f(x) has a...
relative minimum
critical points and endpoints
two different types of functions are multiplied
point of inflection

43. Area between two curves
e^x
uv' + vu'
Area of trapezoid
? f(x) - g(x) over interval a to b - where f(x) is top function and g(x) is bottom function

44. When f '(x) is increasing - f(x) is...
lim as n approaches zero of general term = 0 and terms decrease - series converges
concave up
velocity is negative
Limit as h approaches 0 of [f(a+h)-f(a)]/h

45. Find interval of convergence
use ratio test - set > 1 and solve absolute value equations - check endpoints
? abs[v(t)] over interval a to b
if lim as n approaches 8 of ratio of comparison series/general term is positive and finite - then series behaves like comparison series
find first derivative - dy/dx = dy/dt / dx/dt - then find derivative of first derivative - then divide by dx/dt

46. Linearization
use tangent line to approximate values of the function
negative
0/0 - 8/8 - 8*0 - 8 - 8 - 1^8 - 0° - 8°
Slope of secant line between two points - use to estimate instantanous rate of change at a point.

47. right riemann sum
y' = -sin(x)
? v (dx/dt)² + (dy/dt)² over interval from a to b
uv' + vu'
use rectangles with right-endpoints to evaluate integrals (estimate area)

48. Eatio test
Alternating series converges and general term diverges with another test
lim as n approaches 8 of ratio of (n+1) term/nth term > 1 - series converges
y' = -1/v(1 - x²)
1/2 ? R² - r² over interval from a to b - find a & b by setting equations equal - solve for theta.

49. Instantenous Rate of Change
velocity is negative
logistic growth equation
Slope of tangent line at a point - value of derivative at a point
uv' + vu'

50. [(h1 - h2)/2]*base
? v(1 + (dy/dx)²) dx over interval a to b
use rectangles with right-endpoints to evaluate integrals (estimate area)
Area of trapezoid
Alternating series converges and general term converges with another test