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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. Fundamental Theorem of Calculus
concave up
uv - ? v du
undefined
? f(x) dx on interval a to b = F(b) - F(a)

2. To draw a slope field - plug (x -y) coordinates into differential equation...
if terms grow without bound - series diverges
? f(x) dx on interval a to b = F(b) - F(a)
draw short segments representing slope at each point
no limits - find antiderivative + C - use inital value to find C

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

4. Alternate definition of derivative
if integral converges - series converges
Limit as x approaches a of [f(x)-f(a)]/(x-a)
positive
? v (dx/dt)² + (dy/dt)² over interval from a to b

5. When f '(x) is negative - f(x) is...
decreasing
chain rule
polynomial with finite number of terms - largest exponent is 6 - find all derivatives up to the 6th derivative
concave down

6. Find radius of convergence
relative maximum
? A(x) dx over interval a to b - where A(x) is the area of the given cross-section in terms of x
if lim as n approaches 8 of ratio of comparison series/general term is positive and finite - then series behaves like comparison series
use ratio test - set > 1 and solve absolute value equations - radius = center - endpoint

7. left riemann sum
use rectangles with right-endpoints to evaluate integrals (estimate area)
negative
use rectangles with left-endpoints to evaluate integral (estimate area)
logistic differential equation - M = carrying capacity

8. Converges absolutely
concave down
? v (dx/dt)² + (dy/dt)² over interval from a to b
Alternating series converges and general term converges with another test
1/(b-a) ? f(x) dx on interval a to b

9. Particle is moving to the left/down
velocity is negative
lim as n approaches zero of general term = 0 and terms decrease - series converges
y' = cos(x)
if lim as n approaches 8 of ratio of comparison series/general term is positive and finite - then series behaves like comparison series

10. y = ln(x)/x² - state rule used to find derivative
f '(g(x)) g'(x)
if terms grow without bound - series diverges
quotient rule
uv - ? v du

11. slope of vertical line
velocity is negative
undefined
polynomial with finite number of terms - largest exponent is 6 - find all derivatives up to the 6th derivative
use tangent line to approximate values of the function

12. Particle is moving to the right/up
Limit as x approaches a of [f(x)-f(a)]/(x-a)
velocity is positive
? f(x) dx integrate over interval a to b
Area of trapezoid

13. y = sin?¹(x) - y' =

14. Formal definition of derivative
increasing
Limit as h approaches 0 of [f(a+h)-f(a)]/h
general term = 1/n^p - converges if p > 1
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

15. nth term test
f(x) has a relative minimum
chain rule
if terms grow without bound - series diverges
two different types of functions are multiplied

16. ? u dv =
1/(b-a) ? f(x) dx on interval a to b
e^x
uv - ? v du
? v (dx/dt)² + (dy/dt)² over interval from a to b

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

18. y = cot(x) - y' =

19. y = cos?¹(x) - y' =

20. y = a^x - y' = y' =
speed
a^x ln(a)
? v(1 + (dy/dx)²) dx over interval a to b
corner - cusp - vertical tangent - discontinuity

21. L'Hopitals rule
two different types of functions are multiplied
use to find indeterminate limits - find derivative of numerator and denominator separately then evaluate limit
Alternating series converges and general term converges with another test
relative minimum

22. [(h1 - h2)/2]*base
if lim as n approaches 8 of ratio of comparison series/general term is positive and finite - then series behaves like comparison series
critical points and endpoints
Area of trapezoid
? v (dx/dt)² + (dy/dt)² over interval from a to b

23. When is a function not differentiable
corner - cusp - vertical tangent - discontinuity
derivative
y' = -1/v(1 - x²)
y' = 1/v(1 - x²)

24. y = cot?¹(x) - y' =

25. p-series test
? abs[v(t)] over interval a to b
velocity is negative
general term = 1/n^p - converges if p > 1
product rule

26. Taylor series
use trapezoids to evaluate integrals (estimate area)
polynomial with infinite number of terms - includes general term
? f(x) - g(x) over interval a to b - where f(x) is top function and g(x) is bottom function
if integral converges - series converges

27. Integral test
Limit as h approaches 0 of [f(a+h)-f(a)]/h
if integral converges - series converges
logistic growth equation
? v (dx/dt)² + (dy/dt)² over interval from a to b

28. If f '(x) = 0 and f'(x) > 0 -...
lim as n approaches 8 of ratio of (n+1) term/nth term > 1 - series converges
f(x) has a relative minimum
use to find indeterminate limits - find derivative of numerator and denominator separately then evaluate limit
0/0 - 8/8 - 8*0 - 8 - 8 - 1^8 - 0° - 8°

29. Given velocity vectors dx/dt and dy/dt - find total distance travelled
? 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)
? v (dx/dt)² + (dy/dt)² over interval from a to b
product rule

30. area below x-axis is...
if integral converges - series converges
negative
positive
decreasing

31. Area inside one polar curve and outside another polar curve
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
? f(x) dx on interval a to b = F(b) - F(a)
undefined

32. Intermediate Value Theorem
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)
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.
v(dx/dt)² + (dy/dt)² not an integral!
? A(x) dx over interval a to b - where A(x) is the area of the given cross-section in terms of x

33. Length of curve
if terms grow without bound - series diverges
relative maximum
? v(1 + (dy/dx)²) dx over interval a to b
draw short segments representing slope at each point

34. y = tan?¹(x) - y' =

35. Chain Rule

36. Quotient Rule

37. indefinite integral
Limit as h approaches 0 of [f(a+h)-f(a)]/h
use rectangles with right-endpoints to evaluate integrals (estimate area)
no limits - find antiderivative + C - use inital value to find C
y' = 1/v(1 - x²)

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

39. Average Rate of Change
0/0 - 8/8 - 8*0 - 8 - 8 - 1^8 - 0° - 8°
Slope of secant line between two points - use to estimate instantanous rate of change at a point.
no limits - find antiderivative + C - use inital value to find C
y' = 1/x

40. definite integral
has limits a & b - find antiderivative - F(b) - F(a)
use ratio test - set > 1 and solve absolute value equations - check endpoints
undefined
use to find indeterminate limits - find derivative of numerator and denominator separately then evaluate limit

41. Linearization
use tangent line to approximate values of the function
v(dx/dt)² + (dy/dt)² not an integral!
integrand is a rational function with a factorable denominator
Limit as x approaches a of [f(x)-f(a)]/(x-a)

42. methods of integration
e^x
negative
substitution - parts - partial fractions
f(x) has a relative minimum

43. To find absolute maximum on closed interval [a - b] - you must consider...
critical points and endpoints
A function and it's derivative are in the integrand
use tangent line to approximate values of the function
quotient rule

44. Volume of solid with base in the plane and given cross-section
Limit as h approaches 0 of [f(a+h)-f(a)]/h
? A(x) dx over interval a to b - where A(x) is the area of the given cross-section in terms of x
e^x
critical points and endpoints

45. Volume of solid of revolution - washer
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 - r² over interval from a to b - find a & b by setting equations equal - solve for theta.
? abs[v(t)] over interval a to b
two different types of functions are multiplied

46. When f '(x) changes fro positive to negative - f(x) has a...
use ratio test - set > 1 and solve absolute value equations - check endpoints
p ? r² dx over interval a to b - where r = distance from curve to axis of revolution
point of inflection
relative maximum

47. y = cos²(3x)
logistic growth equation
chain rule
no limits - find antiderivative + C - use inital value to find C
y' = -csc²(x)

48. P = M / (1 + Ae^(-Mkt))
uv' + vu'
f(x) has a relative maximum
logistic growth equation
concave down

49. trapezoidal rule
use tangent line to approximate values of the function
quotient rule
use trapezoids to evaluate integrals (estimate area)
v(dx/dt)² + (dy/dt)² not an integral!

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