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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. 6th degree Taylor Polynomial
lim as n approaches zero of general term = 0 and terms decrease - series converges
if lim as n approaches 8 of ratio of comparison series/general term is positive and finite - then series behaves like comparison series
polynomial with finite number of terms - largest exponent is 6 - find all derivatives up to the 6th derivative
increasing

2. Linearization
quotient rule
relative maximum
Limit as h approaches 0 of [f(a+h)-f(a)]/h
use tangent line to approximate values of the function

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

4. When f '(x) changes from negative to positive - f(x) has a...
relative minimum
draw short segments representing slope at each point
(uv'-vu')/v²
corner - cusp - vertical tangent - discontinuity

5. 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
y' = sec²(x)
y' = 1/(1 + x²)

6. nth term test
general term = a1r^n - converges if -1 < r < 1
use rectangles with right-endpoints to evaluate integrals (estimate area)
if terms grow without bound - series diverges
decreasing

7. When f '(x) is negative - f(x) is...
negative
relative minimum
decreasing
chain rule

8. rate
lim as n approaches 8 of ratio of (n+1) term/nth term > 1 - series converges
derivative
corner - cusp - vertical tangent - discontinuity
decreasing

9. Chain Rule

10. trapezoidal rule
negative
use trapezoids to evaluate integrals (estimate area)
quotient rule
critical points and endpoints

11. y = ln(x)/x² - state rule used to find derivative
quotient rule
undefined
use to find indeterminate limits - find derivative of numerator and denominator separately then evaluate limit
find first derivative - dy/dx = dy/dt / dx/dt - then find derivative of first derivative - then divide by dx/dt

12. Find interval of convergence
Alternating series converges and general term diverges with another test
use trapezoids to evaluate integrals (estimate area)
use ratio test - set > 1 and solve absolute value equations - check endpoints
y' = 1/(1 + x²)

13. mean value theorem

14. y = tan?¹(x) - y' =

15. Area inside one polar curve and outside another polar curve
Limit as h approaches 0 of [f(a+h)-f(a)]/h
positive
use trapezoids to evaluate integrals (estimate area)
1/2 ? R² - r² over interval from a to b - find a & b by setting equations equal - solve for theta.

16. Volume of solid with base in the plane and given cross-section
point of inflection
? A(x) dx over interval a to b - where A(x) is the area of the given cross-section in terms of x
integrand is a rational function with a factorable denominator
draw short segments representing slope at each point

17. indefinite integral
polynomial with finite number of terms - largest exponent is 6 - find all derivatives up to the 6th derivative
no limits - find antiderivative + C - use inital value to find C
y' = 1/(1 + x²)
undefined

18. y = a^x - y' = y' =
y' = sec²(x)
v(dx/dt)² + (dy/dt)² not an integral!
a^x ln(a)
polynomial with finite number of terms - largest exponent is 6 - find all derivatives up to the 6th derivative

19. Area between two curves
1/2 ? R² - r² over interval from a to b - find a & b by setting equations equal - solve for theta.
y' = 1/x
? f(x) - g(x) over interval a to b - where f(x) is top function and g(x) is bottom function
uv - ? v du

20. y = cos²(3x)
concave down
chain rule
use rectangles with right-endpoints to evaluate integrals (estimate area)
decreasing

21. ? u dv =
uv - ? v du
f(x)
y' = -csc²(x)
Slope of tangent line at a point - value of derivative at a point

22. Product Rule

23. When f '(x) is increasing - f(x) is...
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
? v(t) over interval a to b
concave up
positive

24. Average Rate of Change
y' = 1/(x lna)
Slope of secant line between two points - use to estimate instantanous rate of change at a point.
velocity is negative
A function and it's derivative are in the integrand

25. Length of curve
general term = a1r^n - converges if -1 < r < 1
y' = 1/(x lna)
Slope of secant line between two points - use to estimate instantanous rate of change at a point.
? v(1 + (dy/dx)²) dx over interval a to b

26. Volume of solid of revolution - washer
velocity is positive
y' = 1/x
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
two different types of functions are multiplied

27. If f '(x) = 0 and f'(x) > 0 -...
f(x)
f(x) has a relative minimum
0/0 - 8/8 - 8*0 - 8 - 8 - 1^8 - 0° - 8°
? f(x) - g(x) over interval a to b - where f(x) is top function and g(x) is bottom function

28. When f '(x) changes from increasing to decreasing or decreasing to increasing - f(x) has a...
find first derivative - dy/dx = dy/dt / dx/dt - then find derivative of first derivative - then divide by dx/dt
point of inflection
y' = -csc²(x)
speed

29. Instantenous Rate of Change
increasing
Slope of tangent line at a point - value of derivative at a point
? v(t) over interval a to b
lim as n approaches zero of general term = 0 and terms decrease - series converges

30. average value of f(x)
y' = sec(x)tan(x)
1/(b-a) ? f(x) dx on interval a to b
critical points and endpoints
? f(x) dx on interval a to b = F(b) - F(a)

31. Intermediate Value Theorem
relative minimum
p ? r² dx over interval a to b - where r = distance from curve to axis of revolution
use ratio test - set > 1 and solve absolute value equations - check endpoints
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.

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

33. If g(x) = ? f(t) dt on interval 2 to x - then g'(x) =...
f(x)
decreasing
(uv'-vu')/v²
y' = -sin(x)

34. Given velocity vectors dx/dt and dy/dt - find total distance travelled
? v(t) over interval a to b
a^x ln(a)
general term = 1/n^p - converges if p > 1
? v (dx/dt)² + (dy/dt)² over interval from a to b

35. Quotient Rule

36. Volume of solid of revolution - no washer
? f(x) - g(x) over interval a to b - where f(x) is top function and g(x) is bottom function
p ? r² dx over interval a to b - where r = distance from curve to axis of revolution
Slope of secant line between two points - use to estimate instantanous rate of change at a point.
? A(x) dx over interval a to b - where A(x) is the area of the given cross-section in terms of x

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

38. dP/dt = kP(M - P)
logistic differential equation - M = carrying capacity
1/2 ? r² over interval from a to b - find a & b by setting r = 0 - solve for theta
chain rule
decreasing

39. area below x-axis is...
general term = 1/n^p - converges if p > 1
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)
negative
y' = 1/v(1 - x²)

40. Use partial fractions to integrate when...
integrand is a rational function with a factorable denominator
A function and it's derivative are in the integrand
polynomial with infinite number of terms - includes general term
decreasing

41. definite integral
y' = -sin(x)
concave down
has limits a & b - find antiderivative - F(b) - F(a)
Limit as x approaches a of [f(x)-f(a)]/(x-a)

42. Alternating series tes
e^x
product rule
general term = 1/n^p - converges if p > 1
lim as n approaches zero of general term = 0 and terms decrease - series converges

43. Eatio test
Limit as x approaches a of [f(x)-f(a)]/(x-a)
positive
lim as n approaches 8 of ratio of (n+1) term/nth term > 1 - series converges
? f(x) - g(x) over interval a to b - where f(x) is top function and g(x) is bottom function

44. Find radius of convergence
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
use ratio test - set > 1 and solve absolute value equations - radius = center - endpoint
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' = -csc(x)cot(x)

45. Limit comparison test
v(dx/dt)² + (dy/dt)² not an integral!
polynomial with infinite number of terms - includes general term
if lim as n approaches 8 of ratio of comparison series/general term is positive and finite - then series behaves like comparison series
(uv'-vu')/v²

46. methods of integration
speed
quotient rule
y' = 1/x
substitution - parts - partial fractions

47. L'Hopitals rule
no limits - find antiderivative + C - use inital value to find C
y' = -1/v(1 - x²)
use to find indeterminate limits - find derivative of numerator and denominator separately then evaluate limit
? abs[v(t)] over interval a to b

48. use substitution to integrate when

49. Given velocity vectors dx/dt and dy/dt - find speed
use ratio test - set > 1 and solve absolute value equations - check endpoints
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 tangent line to approximate values of the function
v(dx/dt)² + (dy/dt)² not an integral!

50. Area inside polar curve
f(x) has a relative maximum
1/2 ? r² over interval from a to b - find a & b by setting r = 0 - solve for theta
v(dx/dt)² + (dy/dt)² not an integral!
p ? r² dx over interval a to b - where r = distance from curve to axis of revolution