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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 = log (base a) x - y' =

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

3. y = a^x - y' = y' =
a^x ln(a)
zero
increasing
? v (dx/dt)² + (dy/dt)² over interval from a to b

4. y = sin?¹(x) - y' =

5. When f '(x) is increasing - f(x) is...
concave up
use tangent line to approximate values of the function
velocity is negative
? f(x) dx on interval a to b = F(b) - F(a)

6. ? u dv =
uv - ? v du
y' = 1/x
if terms grow without bound - series diverges
y' = -csc²(x)

7. Instantenous Rate of Change
Slope of tangent line at a point - value of derivative at a point
decreasing
(uv'-vu')/v²
? v (dx/dt)² + (dy/dt)² over interval from a to b

8. Find interval of convergence
chain rule
velocity is positive
use ratio test - set > 1 and solve absolute value equations - check endpoints
y' = sec(x)tan(x)

9. Quotient Rule

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

11. p-series test
? v(t) over interval a to b
y' = 1/v(1 - x²)
velocity is negative
general term = 1/n^p - converges if p > 1

12. Converges conditionally
v(dx/dt)² + (dy/dt)² not an integral!
y' = 1/v(1 - x²)
Alternating series converges and general term diverges with another test
lim as n approaches zero of general term = 0 and terms decrease - series converges

13. Limit comparison test
has limits a & b - find antiderivative - F(b) - F(a)
if lim as n approaches 8 of ratio of comparison series/general term is positive and finite - then series behaves like comparison series
negative
point of inflection

14. Volume of solid of revolution - no washer
p ? r² dx over interval a to b - where r = distance from curve to axis of revolution
derivative
point of inflection
separate variables - integrate + C - use initial condition to find C - solve for y

15. y = cos?¹(x) - y' =

16. When f '(x) changes fro positive to negative - f(x) has a...
Limit as x approaches a of [f(x)-f(a)]/(x-a)
y' = -csc(x)cot(x)
relative maximum
1/(b-a) ? f(x) dx on interval a to b

17. Formal definition of derivative
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.
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 h approaches 0 of [f(a+h)-f(a)]/h
f(x) has a relative minimum

18. [(h1 - h2)/2]*base
velocity is positive
Area of trapezoid
y' = -csc²(x)
if terms grow without bound - series diverges

19. y = cot?¹(x) - y' =

20. dP/dt = kP(M - P)
uv' + vu'
separate variables - integrate + C - use initial condition to find C - solve for y
logistic differential equation - M = carrying capacity
y' = 1/x

21. Linearization
? v (dx/dt)² + (dy/dt)² over interval from a to b
? v(t) over interval a to b
use trapezoids to evaluate integrals (estimate area)
use tangent line to approximate values of the function

22. P = M / (1 + Ae^(-Mkt))
p ? r² dx over interval a to b - where r = distance from curve to axis of revolution
increasing
use to find indeterminate limits - find derivative of numerator and denominator separately then evaluate limit
logistic growth equation

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

24. Find radius of convergence
? v(1 + (dy/dx)²) dx over interval a to b
? f(x) dx integrate over interval a to b
use ratio test - set > 1 and solve absolute value equations - radius = center - endpoint
if integral converges - series converges

25. slope of horizontal line
two different types of functions are multiplied
zero
? abs[v(t)] over interval a to b
y' = 1/x

26. area below x-axis is...
negative
concave down
Alternating series converges and general term diverges with another test
p ? r² dx over interval a to b - where r = distance from curve to axis of revolution

27. left riemann sum
y' = sec(x)tan(x)
? v (dx/dt)² + (dy/dt)² over interval from a to b
chain rule
use rectangles with left-endpoints to evaluate integral (estimate area)

28. Area inside one polar curve and outside another polar curve
if integral converges - series converges
1/2 ? R² - r² over interval from a to b - find a & b by setting equations equal - solve for theta.
? f(x) dx integrate over interval a to b
? v(t) over interval a to b

29. 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' = cos(x)
? f(x) dx on interval a to b = F(b) - F(a)
y' = sec(x)tan(x)

30. Product Rule

31. Use partial fractions to integrate when...
integrand is a rational function with a factorable denominator
y' = 1/(x lna)
no limits - find antiderivative + C - use inital value to find C
general term = 1/n^p - converges if p > 1

32. When f '(x) is positive - f(x) is...
p ? r² dx over interval a to b - where r = distance from curve to axis of revolution
logistic growth equation
increasing
product rule

33. Converges absolutely
two different types of functions are multiplied
increasing
Alternating series converges and general term converges with another test
polynomial with finite number of terms - largest exponent is 6 - find all derivatives up to the 6th derivative

34. y = e^x - y' = y' =
v(dx/dt)² + (dy/dt)² not an integral!
f(x) has a relative minimum
? v(1 + (dy/dx)²) dx over interval a to b
e^x

35. Given velocity vectors dx/dt and dy/dt - find total distance travelled
p ? r² dx over interval a to b - where r = distance from curve to axis of revolution
velocity is positive
uv' + vu'
? v (dx/dt)² + (dy/dt)² over interval from a to b

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

37. Average Rate of Change
1/2 ? R² - r² over interval from a to b - find a & b by setting equations equal - solve for theta.
Slope of secant line between two points - use to estimate instantanous rate of change at a point.
y' = -sin(x)
use rectangles with right-endpoints to evaluate integrals (estimate area)

38. methods of integration
Alternating series converges and general term converges with another test
if lim as n approaches 8 of ratio of comparison series/general term is positive and finite - then series behaves like comparison series
substitution - parts - partial fractions
y' = sec(x)tan(x)

39. To find absolute maximum on closed interval [a - b] - you must consider...
critical points and endpoints
f '(g(x)) g'(x)
Slope of tangent line at a point - value of derivative at a point
(uv'-vu')/v²

40. slope of vertical line
decreasing
? v (dx/dt)² + (dy/dt)² over interval from a to b
undefined
draw short segments representing slope at each point

41. definite integral
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
Area of trapezoid
has limits a & b - find antiderivative - F(b) - F(a)
if terms grow without bound - series diverges

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

43. Fundamental Theorem of Calculus
? f(x) dx on interval a to b = F(b) - F(a)
use rectangles with right-endpoints to evaluate integrals (estimate area)
no limits - find antiderivative + C - use inital value to find C
f(x) has a relative maximum

44. Second derivative of parametrically defined curve
? f(x) dx integrate over interval a to b
lim as n approaches zero of general term = 0 and terms decrease - series converges
y' = 1/(1 + x²)
find first derivative - dy/dx = dy/dt / dx/dt - then find derivative of first derivative - then divide by dx/dt

45. To find particular solution to differential equation - dy/dx = x/y...
f(x)
y' = -1/v(1 - x²)
separate variables - integrate + C - use initial condition to find C - solve for y
? A(x) dx over interval a to b - where A(x) is the area of the given cross-section in terms of x

46. Volume of solid of revolution - washer
f(x) has a relative minimum
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
1/2 ? r² 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)

47. Given v(t) find displacement
? v(t) over interval a to b
Limit as x approaches a of [f(x)-f(a)]/(x-a)
y' = cos(x)
? f(x) - g(x) over interval a to b - where f(x) is top function and g(x) is bottom function

48. average value of f(x)
1/(b-a) ? f(x) dx on interval a to b
Limit as x approaches a of [f(x)-f(a)]/(x-a)
increasing
f '(g(x)) g'(x)

49. Particle is moving to the left/down
y' = -csc(x)cot(x)
find first derivative - dy/dx = dy/dt / dx/dt - then find derivative of first derivative - then divide by dx/dt
velocity is negative
if integral converges - series converges

50. Area between two curves
? 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
no limits - find antiderivative + C - use inital value to find C
positive