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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
logistic differential equation - M = carrying capacity
polynomial with finite number of terms - largest exponent is 6 - find all derivatives up to the 6th derivative
find first derivative - dy/dx = dy/dt / dx/dt - then find derivative of first derivative - then divide by dx/dt
y' = -csc²(x)

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

3. Given v(t) find total distance travelled
negative
use ratio test - set > 1 and solve absolute value equations - check endpoints
? abs[v(t)] over interval a to b
y' = 1/x

4. Quotient Rule

5. ? u dv =
uv - ? v du
y' = -csc²(x)
negative
? v(t) over interval a to b

6. If g(x) = ? f(t) dt on interval 2 to x - then g'(x) =...
y' = -1/v(1 - x²)
Limit as h approaches 0 of [f(a+h)-f(a)]/h
f(x)
y' = 1/v(1 - x²)

7. y = cot?¹(x) - y' =

8. [(h1 - h2)/2]*base
y' = 1/x
integrand is a rational function with a factorable denominator
Area of trapezoid
? v (dx/dt)² + (dy/dt)² over interval from a to b

9. L'Hopitals rule
0/0 - 8/8 - 8*0 - 8 - 8 - 1^8 - 0° - 8°
a^x ln(a)
use tangent line to approximate values of the function
use to find indeterminate limits - find derivative of numerator and denominator separately then evaluate limit

10. Formal definition of derivative
if lim as n approaches 8 of ratio of comparison series/general term is positive and finite - then series behaves like comparison series
Limit as h approaches 0 of [f(a+h)-f(a)]/h
velocity is negative
integrand is a rational function with a factorable denominator

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

12. Alternate definition of derivative
undefined
Limit as x approaches a of [f(x)-f(a)]/(x-a)
y' = 1/(x lna)
? v(1 + (dy/dx)²) dx over interval a to b

13. Volume of solid of revolution - no washer
? v(1 + (dy/dx)²) dx over interval a to b
polynomial with infinite number of terms - includes general term
use trapezoids to evaluate integrals (estimate area)
p ? r² dx over interval a to b - where r = distance from curve to axis of revolution

14. absolute value of velocity
point of inflection
? v(1 + (dy/dx)²) dx over interval a to b
speed
y' = 1/x

15. y = sin?¹(x) - y' =

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

17. slope of vertical line
f '(g(x)) g'(x)
undefined
concave up
general term = 1/n^p - converges if p > 1

18. left riemann sum
speed
lim as n approaches 8 of ratio of (n+1) term/nth term > 1 - series converges
derivative
use rectangles with left-endpoints to evaluate integral (estimate area)

19. Chain Rule

20. When f '(x) is positive - f(x) is...
Alternating series converges and general term diverges with another test
y' = -csc²(x)
increasing
(uv'-vu')/v²

21. 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.
chain rule
concave down

22. Given v(t) find displacement
? v(t) over interval a to b
separate variables - integrate + C - use initial condition to find C - solve for y
? v (dx/dt)² + (dy/dt)² over interval from a to b
y' = -sin(x)

23. Use partial fractions to integrate when...
? v (dx/dt)² + (dy/dt)² over interval from a to b
point of inflection
integrand is a rational function with a factorable denominator
quotient rule

24. To find particular solution to differential equation - dy/dx = x/y...
y' = -csc(x)cot(x)
separate variables - integrate + C - use initial condition to find C - solve for y
Limit as x approaches a of [f(x)-f(a)]/(x-a)
y' = -1/v(1 - x²)

25. Indeterminate forms
use trapezoids to evaluate integrals (estimate area)
0/0 - 8/8 - 8*0 - 8 - 8 - 1^8 - 0° - 8°
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.
polynomial with infinite number of terms - includes general term

26. average value of f(x)
1/(b-a) ? f(x) dx on interval a to b
use ratio test - set > 1 and solve absolute value equations - radius = center - endpoint
lim as n approaches zero of general term = 0 and terms decrease - series converges
? v(1 + (dy/dx)²) dx over interval a to b

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

28. When is a function not differentiable
speed
Slope of tangent line at a point - value of derivative at a point
use ratio test - set > 1 and solve absolute value equations - check endpoints
corner - cusp - vertical tangent - discontinuity

29. Taylor series
polynomial with infinite number of terms - includes general term
y' = sec(x)tan(x)
y' = 1/(1 + x²)
1/2 ? r² over interval from a to b - find a & b by setting r = 0 - solve for theta

30. use integration by parts when...
speed
p ? r² dx over interval a to b - where r = distance from curve to axis of revolution
two different types of functions are multiplied
if terms grow without bound - series diverges

31. Average Rate of Change
speed
positive
Slope of secant line between two points - use to estimate instantanous rate of change at a point.
y' = 1/x

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

33. Eatio test
speed
lim as n approaches 8 of ratio of (n+1) term/nth term > 1 - series converges
y' = -1/v(1 - x²)
derivative

34. Converges conditionally
y' = 1/x
y' = -csc(x)cot(x)
y' = -sin(x)
Alternating series converges and general term diverges with another test

35. right riemann sum
if lim as n approaches 8 of ratio of comparison series/general term is positive and finite - then series behaves like comparison series
relative maximum
f(x)
use rectangles with right-endpoints to evaluate integrals (estimate area)

36. mean value theorem

37. Find radius of convergence
decreasing
if lim as n approaches 8 of ratio of comparison series/general term is positive and finite - then series behaves like comparison series
draw short segments representing slope at each point
use ratio test - set > 1 and solve absolute value equations - radius = center - endpoint

38. area below x-axis is...
y' = cos(x)
increasing
point of inflection
negative

39. Instantenous Rate of Change
Slope of tangent line at a point - value of derivative at a point
draw short segments representing slope at each point
integrand is a rational function with a factorable denominator
negative

40. Intermediate Value Theorem
no limits - find antiderivative + C - use inital value to find C
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.
find first derivative - dy/dx = dy/dt / dx/dt - then find derivative of first derivative - then divide by dx/dt
y' = 1/x

41. trapezoidal rule
use trapezoids to evaluate integrals (estimate area)
product rule
Slope of tangent line at a point - value of derivative at a point
if lim as n approaches 8 of ratio of comparison series/general term is positive and finite - then series behaves like comparison series

42. Geometric series test
relative maximum
f(x) has a relative maximum
general term = a1r^n - converges if -1 < r < 1
if lim as n approaches 8 of ratio of comparison series/general term is positive and finite - then series behaves like comparison series

43. When f '(x) is decreasing - f(x) is...
Slope of secant line between two points - use to estimate instantanous rate of change at a point.
concave down
? f(x) - g(x) over interval a to b - where f(x) is top function and g(x) is bottom function
Limit as h approaches 0 of [f(a+h)-f(a)]/h

44. Limit comparison test
use rectangles with right-endpoints to evaluate integrals (estimate area)
corner - cusp - vertical tangent - discontinuity
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/(1 + x²)

45. Particle is moving to the left/down
two different types of functions are multiplied
velocity is negative
uv' + vu'
A function and it's derivative are in the integrand

46. Fundamental Theorem of Calculus
use rectangles with right-endpoints to evaluate integrals (estimate area)
zero
velocity is negative
? f(x) dx on interval a to b = F(b) - F(a)

47. When f '(x) changes from negative to positive - f(x) has a...
relative minimum
use rectangles with left-endpoints to evaluate integral (estimate area)
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.
critical points and endpoints

48. Area between two curves
use trapezoids to evaluate integrals (estimate area)
? f(x) - g(x) over interval a to b - where f(x) is top function and g(x) is bottom function
f(x) has a relative minimum
lim as n approaches 8 of ratio of (n+1) term/nth term > 1 - series converges

49. Volume of solid with base in the plane and given cross-section
? A(x) dx over interval a to b - where A(x) is the area of the given cross-section in terms of x
velocity is negative
y' = -csc²(x)
a^x ln(a)

50. When f '(x) changes from increasing to decreasing or decreasing to increasing - f(x) has a...
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
point of inflection
Limit as x approaches a of [f(x)-f(a)]/(x-a)
decreasing