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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. When f '(x) changes from negative to positive - f(x) has a...
y' = 1/x
undefined
use tangent line to approximate values of the function
relative minimum

2. When f '(x) is increasing - f(x) is...
Alternating series converges and general term converges with another test
concave up
zero
y' = 1/(x lna)

3. rate
Alternating series converges and general term diverges with another test
derivative
logistic growth equation
no limits - find antiderivative + C - use inital value to find C

4. y = sin?¹(x) - y' =

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

6. To find absolute maximum on closed interval [a - b] - you must consider...
1/2 ? R² - r² over interval from a to b - find a & b by setting equations equal - solve for theta.
f(x) has a relative minimum
critical points and endpoints
use rectangles with right-endpoints to evaluate integrals (estimate area)

7. dP/dt = kP(M - P)
1/2 ? r² over interval from a to b - find a & b by setting r = 0 - solve for theta
logistic differential equation - M = carrying capacity
corner - cusp - vertical tangent - discontinuity
has limits a & b - find antiderivative - F(b) - F(a)

8. Find radius of convergence
y' = -csc(x)cot(x)
general term = 1/n^p - converges if p > 1
y' = 1/x
use ratio test - set > 1 and solve absolute value equations - radius = center - endpoint

9. Instantenous Rate of Change
product rule
Area of trapezoid
? v(t) over interval a to b
Slope of tangent line at a point - value of derivative at a point

10. Particle is moving to the left/down
corner - cusp - vertical tangent - discontinuity
if integral converges - series converges
velocity is negative
two different types of functions are multiplied

11. Given velocity vectors dx/dt and dy/dt - find total distance travelled
uv - ? v du
y' = -csc(x)cot(x)
? v (dx/dt)² + (dy/dt)² over interval from a to b
concave down

12. Given v(t) find displacement
? v(t) over interval a to b
use rectangles with right-endpoints to evaluate integrals (estimate area)
Slope of tangent line at a point - value of derivative at a point
polynomial with infinite number of terms - includes general term

13. slope of horizontal line
zero
if integral converges - series converges
1/(b-a) ? f(x) dx on interval a to b
Limit as x approaches a of [f(x)-f(a)]/(x-a)

14. absolute value of velocity
integrand is a rational function with a factorable denominator
Alternating series converges and general term converges with another test
y' = 1/x
speed

15. 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
Alternating series converges and general term converges with another test
p ? r² dx over interval a to b - where r = distance from curve to axis of revolution
use rectangles with right-endpoints to evaluate integrals (estimate area)

16. Taylor series
y' = -csc²(x)
polynomial with infinite number of terms - includes general term
general term = 1/n^p - converges if p > 1
general term = a1r^n - converges if -1 < r < 1

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

18. Area inside polar curve
1/2 ? R² - r² over interval from a to b - find a & b by setting equations equal - solve for theta.
1/2 ? r² over interval from a to b - find a & b by setting r = 0 - solve for theta
critical points and endpoints
uv' + vu'

19. y = x cos(x) - state rule used to find derivative
lim as n approaches zero of general term = 0 and terms decrease - series converges
critical points and endpoints
Slope of secant line between two points - use to estimate instantanous rate of change at a point.
product rule

20. When is a function not differentiable
corner - cusp - vertical tangent - discontinuity
(uv'-vu')/v²
if lim as n approaches 8 of ratio of comparison series/general term is positive and finite - then series behaves like comparison series
point of inflection

21. slope of vertical line
p ? r² dx over interval a to b - where r = distance from curve to axis of revolution
use tangent line to approximate values of the function
undefined
concave down

22. y = cot?¹(x) - y' =

23. area below x-axis is...
Alternating series converges and general term converges with another test
general term = a1r^n - converges if -1 < r < 1
negative
Slope of tangent line at a point - value of derivative at a point

24. Converges absolutely
concave up
Alternating series converges and general term converges with another test
use to find indeterminate limits - find derivative of numerator and denominator separately then evaluate limit
Limit as h approaches 0 of [f(a+h)-f(a)]/h

25. left riemann sum
use rectangles with left-endpoints to evaluate integral (estimate area)
a^x ln(a)
general term = a1r^n - converges if -1 < r < 1
Slope of tangent line at a point - value of derivative at a point

26. Linearization
use tangent line to approximate values of the function
? f(x) dx integrate over interval a to b
y' = -csc²(x)
1/(b-a) ? f(x) dx on interval a to b

27. Limit comparison test
y' = 1/x
if lim as n approaches 8 of ratio of comparison series/general term is positive and finite - then series behaves like comparison series
velocity is negative
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.

28. right riemann sum
quotient rule
general term = a1r^n - converges if -1 < r < 1
general term = 1/n^p - converges if p > 1
use rectangles with right-endpoints to evaluate integrals (estimate area)

29. indefinite integral
no limits - find antiderivative + C - use inital value to find C
has limits a & b - find antiderivative - F(b) - F(a)
uv - ? v du
zero

30. When f '(x) is positive - f(x) is...
? f(x) dx on interval a to b = F(b) - F(a)
increasing
a^x ln(a)
positive

31. mean value theorem

32. Volume of solid of revolution - washer
? v(1 + (dy/dx)²) dx over interval a to b
Limit as x approaches a of [f(x)-f(a)]/(x-a)
use rectangles with right-endpoints 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

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

34. area under a curve
? f(x) dx integrate over interval a to b
point of inflection
uv' + vu'
use to find indeterminate limits - find derivative of numerator and denominator separately then evaluate limit

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

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

37. When f '(x) is negative - f(x) is...
decreasing
v(dx/dt)² + (dy/dt)² not an integral!
y' = sec(x)tan(x)
e^x

38. Particle is moving to the right/up
use trapezoids to evaluate integrals (estimate area)
draw short segments representing slope at each point
velocity is positive
? v (dx/dt)² + (dy/dt)² over interval from a to b

39. Given velocity vectors dx/dt and dy/dt - find speed
v(dx/dt)² + (dy/dt)² not an integral!
use rectangles with right-endpoints to evaluate integrals (estimate area)
f '(g(x)) g'(x)
zero

40. Alternate definition of derivative
Limit as x approaches a of [f(x)-f(a)]/(x-a)
lim as n approaches 8 of ratio of (n+1) term/nth term > 1 - series converges
if integral converges - series converges
general term = 1/n^p - converges if p > 1

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

42. Volume of solid of revolution - no washer
speed
concave up
? f(x) dx integrate over interval a to b
p ? r² dx over interval a to b - where r = distance from curve to axis of revolution

43. When f '(x) changes from increasing to decreasing or decreasing to increasing - f(x) has a...
Limit as x approaches a of [f(x)-f(a)]/(x-a)
point of inflection
? v (dx/dt)² + (dy/dt)² over interval from a to b
has limits a & b - find antiderivative - F(b) - F(a)

44. Product Rule

45. use integration by parts when...
y' = -1/v(1 - x²)
(uv'-vu')/v²
two different types of functions are multiplied
use ratio test - set > 1 and solve absolute value equations - check endpoints

46. Integral test
? f(x) - g(x) over interval a to b - where f(x) is top function and g(x) is bottom function
increasing
use ratio test - set > 1 and solve absolute value equations - check endpoints
if integral converges - series converges

47. y = a^x - y' = y' =
y' = -sin(x)
use tangent line to approximate values of the function
general term = 1/n^p - converges if p > 1
a^x ln(a)

48. Second derivative of parametrically defined curve
? f(x) - g(x) over interval a to b - where f(x) is top function and g(x) is bottom function
product rule
y' = -csc²(x)
find first derivative - dy/dx = dy/dt / dx/dt - then find derivative of first derivative - then divide by dx/dt

49. Alternating series tes
? v(t) over interval a to b
zero
lim as n approaches zero of general term = 0 and terms decrease - series converges
1/(b-a) ? f(x) dx on interval a to b

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