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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. y = x cos(x) - state rule used to find derivative
product rule
(uv'-vu')/v²
zero
find first derivative - dy/dx = dy/dt / dx/dt - then find derivative of first derivative - then divide by dx/dt

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

3. slope of horizontal line
product rule
y' = -csc²(x)
zero
? f(x) dx integrate over interval a to b

4. Indeterminate forms
y' = cos(x)
undefined
0/0 - 8/8 - 8*0 - 8 - 8 - 1^8 - 0° - 8°
velocity is negative

5. area below x-axis is...
decreasing
integrand is a rational function with a factorable denominator
? f(x) dx integrate over interval a to b
negative

6. Area between two curves
y' = 1/v(1 - x²)
quotient rule
? f(x) - g(x) over interval a to b - where f(x) is top function and g(x) is bottom function
y' = cos(x)

7. Length of curve
speed
? f(x) dx integrate over interval a to b
? v(1 + (dy/dx)²) dx over interval a to b
if integral converges - series converges

8. Linearization
polynomial with finite number of terms - largest exponent is 6 - find all derivatives up to the 6th derivative
undefined
use tangent line to approximate values of the function
y' = sec(x)tan(x)

9. Taylor series
use rectangles with left-endpoints to evaluate integral (estimate area)
no limits - find antiderivative + C - use inital value to find C
? abs[v(t)] over interval a to b
polynomial with infinite number of terms - includes general term

10. When f '(x) changes fro positive to negative - f(x) has 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.
point of inflection
relative maximum
use rectangles with right-endpoints to evaluate integrals (estimate area)

11. Given velocity vectors dx/dt and dy/dt - find total distance travelled
chain rule
draw short segments representing slope at each point
? v (dx/dt)² + (dy/dt)² over interval from a to b
y' = 1/(1 + x²)

12. definite integral
positive
y' = -1/v(1 - x²)
lim as n approaches 8 of ratio of (n+1) term/nth term > 1 - series converges
has limits a & b - find antiderivative - F(b) - F(a)

13. Alternating series tes
polynomial with finite number of terms - largest exponent is 6 - find all derivatives up to the 6th derivative
lim as n approaches zero of general term = 0 and terms decrease - series converges
? f(x) dx on interval a to b = F(b) - F(a)
draw short segments representing slope at each point

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

15. Area inside polar curve
negative
1/2 ? r² over interval from a to b - find a & b by setting r = 0 - solve for theta
if integral converges - series converges
concave up

16. Given velocity vectors dx/dt and dy/dt - find speed
y' = 1/v(1 - x²)
use rectangles with right-endpoints to evaluate integrals (estimate area)
v(dx/dt)² + (dy/dt)² not an integral!
relative minimum

17. Find interval of convergence
use ratio test - set > 1 and solve absolute value equations - check endpoints
f(x)
if terms grow without bound - series diverges
no limits - find antiderivative + C - use inital value to find C

18. p-series test
y' = -csc(x)cot(x)
logistic differential equation - M = carrying capacity
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
general term = 1/n^p - converges if p > 1

19. y = e^x - y' = y' =
y' = -sin(x)
e^x
uv - ? v du
use ratio test - set > 1 and solve absolute value equations - check endpoints

20. y = log (base a) x - y' =

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

22. Given v(t) find total distance travelled
? abs[v(t)] over interval a to b
0/0 - 8/8 - 8*0 - 8 - 8 - 1^8 - 0° - 8°
lim as n approaches 8 of ratio of (n+1) term/nth term > 1 - series converges
speed

23. Length of parametric curve
product rule
Limit as h approaches 0 of [f(a+h)-f(a)]/h
? 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

24. Particle is moving to the right/up
velocity is positive
use ratio test - set > 1 and solve absolute value equations - check endpoints
critical points and endpoints
corner - cusp - vertical tangent - discontinuity

25. Particle is moving to the left/down
velocity is negative
a^x ln(a)
logistic growth equation
? v (dx/dt)² + (dy/dt)² over interval from a to b

26. When f '(x) is negative - f(x) is...
y' = sec²(x)
decreasing
e^x
? f(x) dx on interval a to b = F(b) - F(a)

27. Alternate definition of derivative
Limit as x approaches a of [f(x)-f(a)]/(x-a)
general term = 1/n^p - converges if p > 1
draw short segments representing slope at each point
? abs[v(t)] over interval a to b

28. left riemann sum
use rectangles with right-endpoints to evaluate integrals (estimate area)
separate variables - integrate + C - use initial condition to find C - solve for y
y' = 1/(x lna)
use rectangles with left-endpoints to evaluate integral (estimate area)

29. y = a^x - y' = y' =
polynomial with infinite number of terms - includes general term
has limits a & b - find antiderivative - F(b) - F(a)
a^x ln(a)
? f(x) dx on interval a to b = F(b) - F(a)

30. Limit comparison test
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) - g(x) over interval a to b - where f(x) is top function and g(x) is bottom function
find first derivative - dy/dx = dy/dt / dx/dt - then find derivative of first derivative - then divide by dx/dt
Alternating series converges and general term diverges with another test

31. Area inside one polar curve and outside another polar curve
decreasing
zero
1/2 ? R² - r² over interval from a to b - find a & b by setting equations equal - solve for theta.
y' = 1/(1 + x²)

32. Integral test
? v(1 + (dy/dx)²) dx over interval a to b
if integral converges - series converges
relative minimum
positive

33. area under a curve
uv' + vu'
two different types of functions are multiplied
Area of trapezoid
? f(x) dx integrate over interval a to b

34. indefinite integral
y' = 1/(1 + x²)
no limits - find antiderivative + C - use inital value to find C
point of inflection
increasing

35. y = tan?¹(x) - y' =

36. Volume of solid of revolution - no washer
p ? r² dx over interval a to b - where r = distance from curve to axis of revolution
1/2 ? r² over interval from a to b - find a & b by setting r = 0 - solve for theta
if integral converges - series converges
? v (dx/dt)² + (dy/dt)² over interval from a to b

37. Product Rule

38. use substitution to integrate when

39. average value of f(x)
? v(t) over interval a to b
1/(b-a) ? f(x) dx on interval a to b
? f(x) dx integrate over interval a to b
two different types of functions are multiplied

40. When f '(x) is positive - f(x) is...
Slope of tangent line at a point - value of derivative at a point
y' = -csc²(x)
relative minimum
increasing

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

42. L'Hopitals rule
point of inflection
use to find indeterminate limits - find derivative of numerator and denominator separately then evaluate limit
two different types of functions are multiplied
1/2 ? R² - r² over interval from a to b - find a & b by setting equations equal - solve for theta.

43. If f '(x) = 0 and f'(x) > 0 -...
draw short segments representing slope at each point
two different types of functions are multiplied
f(x) has a relative minimum
if lim as n approaches 8 of ratio of comparison series/general term is positive and finite - then series behaves like comparison series

44. absolute value of velocity
? v(1 + (dy/dx)²) dx over interval a to b
use ratio test - set > 1 and solve absolute value equations - check endpoints
speed
y' = cos(x)

45. When f '(x) changes from increasing to decreasing or decreasing to increasing - f(x) has a...
point of inflection
Limit as h approaches 0 of [f(a+h)-f(a)]/h
negative
logistic differential equation - M = carrying capacity

46. If f '(x) = 0 and f'(x) < 0 -...
velocity is negative
y' = 1/x
undefined
f(x) has a relative maximum

47. right riemann sum
concave down
? 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)
use rectangles with right-endpoints to evaluate integrals (estimate area)

48. P = M / (1 + Ae^(-Mkt))
zero
y' = sec²(x)
logistic growth equation
y' = 1/x

49. dP/dt = kP(M - P)
logistic differential equation - M = carrying capacity
use ratio test - set > 1 and solve absolute value equations - check endpoints
Limit as h approaches 0 of [f(a+h)-f(a)]/h
velocity is negative

50. Quotient Rule