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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. To draw a slope field - plug (x -y) coordinates into differential equation...
draw short segments representing slope at each point
polynomial with infinite number of terms - includes general term
? abs[v(t)] over interval a to b
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)

2. Given velocity vectors dx/dt and dy/dt - find total distance travelled
undefined
use rectangles with left-endpoints to evaluate integral (estimate area)
polynomial with infinite number of terms - includes general term
? v (dx/dt) + (dy/dt) over interval from a to b

3. y = log (base a) x - y' =
use trapezoids to evaluate integrals (estimate area)
lim as n approaches 8 of ratio of (n+1) term/nth term > 1 - series converges
y' = 1/v(1 - x)
y' = 1/(x lna)

4. [(h1 - h2)/2]*base
Slope of secant line between two points - use to estimate instantanous rate of change at a point.
Area of trapezoid
Alternating series converges and general term diverges with another test
Alternating series converges and general term converges with another test

5. Indeterminate forms
concave up
0/0 - 8/8 - 8*0 - 8 - 8 - 1^8 - 0 - 8
decreasing
zero

6. dP/dt = kP(M - P)
chain rule
product rule
Limit as x approaches a of [f(x)-f(a)]/(x-a)
logistic differential equation - M = carrying capacity

7. definite integral
? f(x) dx integrate over interval a to b
? v (dx/dt) + (dy/dt) over interval from a to b
product rule
has limits a & b - find antiderivative - F(b) - F(a)

8. trapezoidal rule
? f(x) - g(x) over interval a to b - where f(x) is top function and g(x) is bottom function
f '(g(x)) g'(x)
two different types of functions are multiplied
use trapezoids to evaluate integrals (estimate area)

9. y = cot?(x) - y' =
? f(x) - g(x) over interval a to b - where f(x) is top function and g(x) is bottom function
1/(b-a) ? f(x) dx on interval a to b
speed
y' = -1/(1 + x)

10. When f '(x) changes from negative to positive - f(x) has a...
? v(1 + (dy/dx)) dx over interval a to b
negative
? v(t) over interval a to b
relative minimum

11. When f '(x) is decreasing - f(x) is...
decreasing
1/(b-a) ? f(x) dx on interval a to b
concave down
use ratio test - set > 1 and solve absolute value equations - check endpoints

12. right riemann sum
y' = -1/v(1 - x)
use rectangles with right-endpoints to evaluate integrals (estimate area)
quotient rule
general term = 1/n^p - converges if p > 1

13. Given velocity vectors dx/dt and dy/dt - find speed
decreasing
positive
v(dx/dt) + (dy/dt) not an integral!
derivative

14. To find absolute maximum on closed interval [a - b] - you must consider...
critical points and endpoints
use trapezoids to evaluate integrals (estimate area)
? v (dx/dt) + (dy/dt) over interval from 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

15. If f '(x) = 0 and f'(x) > 0 -...
undefined
use rectangles with right-endpoints to evaluate integrals (estimate area)
? v(t) over interval a to b
f(x) has a relative minimum

16. area below x-axis is...
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)
? A(x) dx over interval a to b - where A(x) is the area of the given cross-section in terms of x
logistic differential equation - M = carrying capacity
negative

17. indefinite integral
y' = 1/v(1 - x)
point of inflection
uv' + vu'
no limits - find antiderivative + C - use inital value to find C

18. Converges absolutely
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.
Alternating series converges and general term converges with another test
corner - cusp - vertical tangent - discontinuity
? A(x) dx over interval a to b - where A(x) is the area of the given cross-section in terms of x

19. When f '(x) changes from increasing to decreasing or decreasing to increasing - f(x) has a...
? v (dx/dt) + (dy/dt) over interval from a to b
point of inflection
y' = 1/(1 + x)
chain rule

20. If f '(x) = 0 and f'(x) < 0 -...
increasing
f(x) has a relative maximum
derivative
1/2 ? R - r over interval from a to b - find a & b by setting equations equal - solve for theta.

21. Product Rule
concave up
uv' + vu'
find first derivative - dy/dx = dy/dt / dx/dt - then find derivative of first derivative - then divide by dx/dt
use rectangles with right-endpoints to evaluate integrals (estimate area)

22. P = M / (1 + Ae^(-Mkt))
y' = sec(x)
velocity is positive
logistic growth equation
derivative

23. Instantenous Rate of Change
positive
Slope of tangent line at a point - value of derivative at a point
chain rule
integrand is a rational function with a factorable denominator

24. slope of vertical line
use to find indeterminate limits - find derivative of numerator and denominator separately then evaluate limit
y' = -csc(x)cot(x)
undefined
speed

25. Area between two curves
a^x ln(a)
polynomial with infinite number of terms - includes general term
A function and it's derivative are in the integrand
? f(x) - g(x) over interval a to b - where f(x) is top function and g(x) is bottom function

26. ? u dv =
uv - ? v du
velocity is positive
logistic growth equation
f '(g(x)) g'(x)

27. Find interval of convergence
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)
use rectangles with right-endpoints to evaluate integrals (estimate area)
use ratio test - set > 1 and solve absolute value equations - check endpoints
if terms grow without bound - series diverges

28. y = sec(x) - y' =
velocity is positive
Slope of secant line between two points - use to estimate instantanous rate of change at a point.
y' = sec(x)tan(x)
1/(b-a) ? f(x) dx on interval a to b

29. Particle is moving to the left/down
has limits a & b - find antiderivative - F(b) - F(a)
velocity is negative
lim as n approaches 8 of ratio of (n+1) term/nth term > 1 - series converges
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)

30. If g(x) = ? f(t) dt on interval 2 to x - then g'(x) =...
f(x)
(uv'-vu')/v
quotient rule
use to find indeterminate limits - find derivative of numerator and denominator separately then evaluate limit

31. 6th degree Taylor Polynomial
y' = -csc(x)
polynomial with finite number of terms - largest exponent is 6 - find all derivatives up to the 6th derivative
uv - ? v du
use rectangles with right-endpoints to evaluate integrals (estimate area)

32. Use partial fractions to integrate when...
y' = sec(x)tan(x)
f(x) has a relative maximum
Slope of secant line between two points - use to estimate instantanous rate of change at a point.
integrand is a rational function with a factorable denominator

33. Alternating series tes
? v(t) over interval a to b
lim as n approaches zero of general term = 0 and terms decrease - series converges
no limits - find antiderivative + C - use inital value to find C
0/0 - 8/8 - 8*0 - 8 - 8 - 1^8 - 0 - 8

34. p-series test
a^x ln(a)
logistic differential equation - M = carrying capacity
product rule
general term = 1/n^p - converges if p > 1

35. Chain Rule
Limit as x approaches a of [f(x)-f(a)]/(x-a)
f '(g(x)) g'(x)
chain rule
Alternating series converges and general term converges with another test

36. y = cos?(x) - y' =
chain rule
y' = -1/v(1 - x)
0/0 - 8/8 - 8*0 - 8 - 8 - 1^8 - 0 - 8
use rectangles with left-endpoints to evaluate integral (estimate area)

37. Quotient Rule
use ratio test - set > 1 and solve absolute value equations - check endpoints
(uv'-vu')/v
integrand is a rational function with a factorable denominator
draw short segments representing slope at each point

38. 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)
? f(x) dx on interval a to b = F(b) - F(a)
polynomial with finite number of terms - largest exponent is 6 - find all derivatives up to the 6th derivative

39. Second derivative of parametrically defined curve
y' = 1/x
use rectangles with left-endpoints to evaluate integral (estimate area)
find first derivative - dy/dx = dy/dt / dx/dt - then find derivative of first derivative - then divide by dx/dt
f(x) has a relative minimum

40. Integral test
concave down
if integral converges - series converges
f '(g(x)) g'(x)
y' = -sin(x)

41. Formal definition of derivative
Limit as h approaches 0 of [f(a+h)-f(a)]/h
(uv'-vu')/v
? abs[v(t)] over interval a to b
y' = 1/(1 + x)

42. slope of horizontal line
derivative
zero
y' = sec(x)
use rectangles with left-endpoints to evaluate integral (estimate area)

43. Linearization
use tangent line to approximate values of the function
y' = 1/(x lna)
? A(x) dx over interval a to b - where A(x) is the area of the given cross-section in terms of x
a^x ln(a)

44. left riemann sum
use rectangles with left-endpoints to evaluate integral (estimate area)
y' = 1/x
find first derivative - dy/dx = dy/dt / dx/dt - then find derivative of first derivative - then divide by dx/dt
polynomial with finite number of terms - largest exponent is 6 - find all derivatives up to the 6th derivative

45. mean value theorem
? v (dx/dt) + (dy/dt) over interval from a to b
has limits a & b - find antiderivative - F(b) - F(a)
Slope of tangent line at a point - value of derivative at a point
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)

46. Taylor series
? f(x) dx on interval a to b = F(b) - F(a)
polynomial with infinite number of terms - includes general term
y' = cos(x)
polynomial with finite number of terms - largest exponent is 6 - find all derivatives up to the 6th derivative

47. When f '(x) is positive - f(x) is...
a^x ln(a)
speed
increasing
positive

48. When is a function not differentiable
0/0 - 8/8 - 8*0 - 8 - 8 - 1^8 - 0 - 8
corner - cusp - vertical tangent - discontinuity
Alternating series converges and general term converges with another test
two different types of functions are multiplied

49. y = sin(x) - y' =
f(x)
y' = cos(x)
has limits a & b - find antiderivative - F(b) - F(a)
velocity is positive

50. Alternate definition of derivative
a^x ln(a)
y' = sec(x)tan(x)
Limit as x approaches a of [f(x)-f(a)]/(x-a)
substitution - parts - partial fractions

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