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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. mean value theorem

2. trapezoidal rule
use trapezoids to evaluate integrals (estimate area)
Alternating series converges and general term converges with another test
two different types of functions are multiplied
polynomial with finite number of terms - largest exponent is 6 - find all derivatives up to the 6th derivative

3. Given v(t) find total distance travelled
? f(x) dx on interval a to b = F(b) - F(a)
y' = -sin(x)
A function and it's derivative are in the integrand
? abs[v(t)] over interval a to b

4. Integral test
? A(x) dx over interval a to b - where A(x) is the area of the given cross-section in terms of x
if integral converges - series converges
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.
0/0 - 8/8 - 8*0 - 8 - 8 - 1^8 - 0° - 8°

5. y = cos²(3x)
0/0 - 8/8 - 8*0 - 8 - 8 - 1^8 - 0° - 8°
chain rule
f '(g(x)) g'(x)
if integral converges - series converges

6. Chain Rule

7. When f '(x) changes from increasing to decreasing or decreasing to increasing - f(x) has a...
point of inflection
corner - cusp - vertical tangent - discontinuity
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.
velocity is positive

8. indefinite integral
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)
no limits - find antiderivative + C - use inital value to find C
p ? r² dx over interval a to b - where r = distance from curve to axis of revolution
find first derivative - dy/dx = dy/dt / dx/dt - then find derivative of first derivative - then divide by dx/dt

9. When f '(x) is increasing - f(x) is...
relative maximum
find first derivative - dy/dx = dy/dt / dx/dt - then find derivative of first derivative - then divide by dx/dt
0/0 - 8/8 - 8*0 - 8 - 8 - 1^8 - 0° - 8°
concave up

10. y = cos?¹(x) - y' =

11. Indeterminate forms
0/0 - 8/8 - 8*0 - 8 - 8 - 1^8 - 0° - 8°
Area of trapezoid
? v (dx/dt)² + (dy/dt)² over interval from a to b
? v(1 + (dy/dx)²) dx over interval a to b

12. Eatio test
negative
lim as n approaches 8 of ratio of (n+1) term/nth term > 1 - series converges
critical points and endpoints
no limits - find antiderivative + C - use inital value to find C

13. use integration by parts when...
Limit as h approaches 0 of [f(a+h)-f(a)]/h
p ? r² dx over interval a to b - where r = distance from curve to axis of revolution
use to find indeterminate limits - find derivative of numerator and denominator separately then evaluate limit
two different types of functions are multiplied

14. slope of vertical line
undefined
lim as n approaches zero of general term = 0 and terms decrease - series converges
y' = -csc(x)cot(x)
use tangent line to approximate values of the function

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

16. absolute value of velocity
increasing
speed
two different types of functions are multiplied
use ratio test - set > 1 and solve absolute value equations - check endpoints

17. area above x-axis is...
logistic growth equation
e^x
positive
use to find indeterminate limits - find derivative of numerator and denominator separately then evaluate limit

18. Alternate definition of derivative
integrand is a rational function with a factorable denominator
if integral converges - series converges
Limit as x approaches a of [f(x)-f(a)]/(x-a)
polynomial with infinite number of terms - includes general term

19. Quotient Rule

20. If g(x) = ? f(t) dt on interval 2 to x - then g'(x) =...
f(x)
Limit as h approaches 0 of [f(a+h)-f(a)]/h
use tangent line to approximate values of the function
f(x) has a relative minimum

21. L'Hopitals rule
point of inflection
draw short segments representing slope at each point
use to find indeterminate limits - find derivative of numerator and denominator separately then evaluate limit
? v(t) over interval a to b

22. When f '(x) is decreasing - f(x) is...
y' = 1/(x lna)
no limits - find antiderivative + C - use inital value to find C
concave down
separate variables - integrate + C - use initial condition to find C - solve for y

23. Given v(t) find displacement
concave up
A function and it's derivative are in the integrand
? A(x) dx over interval a to b - where A(x) is the area of the given cross-section in terms of x
? v(t) over interval a to b

24. dP/dt = kP(M - P)
logistic differential equation - M = carrying capacity
separate variables - integrate + C - use initial condition to find C - solve for y
if lim as n approaches 8 of ratio of comparison series/general term is positive and finite - then series behaves like comparison series
undefined

25. use substitution to integrate when

26. Use partial fractions to integrate when...
increasing
integrand is a rational function with a factorable denominator
use ratio test - set > 1 and solve absolute value equations - radius = center - endpoint
general term = a1r^n - converges if -1 < r < 1

27. Linearization
? v(t) over interval a to b
decreasing
use tangent line to approximate values of the function
polynomial with infinite number of terms - includes general term

28. Volume of solid of revolution - no washer
y' = 1/(1 + x²)
relative maximum
p ? r² dx over interval a to b - where r = distance from curve to axis of revolution
? v (dx/dt)² + (dy/dt)² over interval from a to b

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

30. p-series test
v(dx/dt)² + (dy/dt)² not an integral!
y' = sec(x)tan(x)
general term = 1/n^p - converges if p > 1
use ratio test - set > 1 and solve absolute value equations - check endpoints

31. Product Rule

32. When f '(x) changes fro positive to negative - f(x) has a...
general term = a1r^n - converges if -1 < r < 1
(uv'-vu')/v²
decreasing
relative maximum

33. right riemann sum
y' = -1/v(1 - x²)
use rectangles with right-endpoints to evaluate integrals (estimate area)
decreasing
lim as n approaches 8 of ratio of (n+1) term/nth term > 1 - series converges

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

35. Length of curve
uv' + vu'
use to find indeterminate limits - find derivative of numerator and denominator separately then evaluate limit
? v(1 + (dy/dx)²) dx over interval a to b
e^x

36. Particle is moving to the left/down
velocity is negative
? f(x) dx integrate over interval a to b
Slope of secant line between two points - use to estimate instantanous rate of change at a point.
negative

37. slope of horizontal line
derivative
zero
Alternating series converges and general term diverges with another test
? abs[v(t)] over interval a to b

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

39. Area inside polar curve
f(x) has a relative minimum
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
Limit as h approaches 0 of [f(a+h)-f(a)]/h

40. 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
if lim as n approaches 8 of ratio of comparison series/general term is positive and finite - then series behaves like comparison series
y' = 1/v(1 - x²)
product rule

41. 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
quotient rule
f(x) has a relative maximum
(uv'-vu')/v²

42. P = M / (1 + Ae^(-Mkt))
logistic growth equation
? v(1 + (dy/dx)²) dx over interval a to b
f(x) has a relative maximum
f '(g(x)) g'(x)

43. area below x-axis is...
y' = sec(x)tan(x)
1/(b-a) ? f(x) dx on interval a to b
e^x
negative

44. Taylor series
polynomial with infinite number of terms - includes general term
substitution - parts - partial fractions
speed
derivative

45. Converges absolutely
if integral converges - series converges
Alternating series converges and general term converges with another test
y' = 1/(1 + x²)
undefined

46. Alternating series tes
lim as n approaches zero of general term = 0 and terms decrease - series converges
? v(t) over interval a to b
e^x
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.

47. Instantenous Rate of Change
polynomial with finite number of terms - largest exponent is 6 - find all derivatives up to the 6th derivative
? v(t) over interval a to b
y' = -csc²(x)
Slope of tangent line at a point - value of derivative at a point

48. When is a function not differentiable
lim as n approaches 8 of ratio of (n+1) term/nth term > 1 - series converges
corner - cusp - vertical tangent - discontinuity
use rectangles with right-endpoints to evaluate integrals (estimate area)
integrand is a rational function with a factorable denominator

49. To find particular solution to differential equation - dy/dx = x/y...
find first derivative - dy/dx = dy/dt / dx/dt - then find derivative of first derivative - then divide by dx/dt
concave down
logistic growth equation
separate variables - integrate + C - use initial condition to find C - solve for y

50. average value of f(x)
velocity is negative
two different types of functions are multiplied
1/(b-a) ? f(x) dx on interval a to b
use ratio test - set > 1 and solve absolute value equations - radius = center - endpoint