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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. dP/dt = kP(M - P)
speed
(uv'-vu')/v²
logistic differential equation - M = carrying capacity
general term = a1r^n - converges if -1 < r < 1

2. Find interval of convergence
integrand is a rational function with a factorable denominator
negative
1/(b-a) ? f(x) dx on interval a to b
use ratio test - set > 1 and solve absolute value equations - check endpoints

3. 6th degree Taylor Polynomial
Alternating series converges and general term diverges with another test
polynomial with finite number of terms - largest exponent is 6 - find all derivatives up to the 6th derivative
general term = 1/n^p - converges if p > 1
separate variables - integrate + C - use initial condition to find C - solve for y

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

5. Area inside one polar curve and outside another polar curve
use rectangles with right-endpoints 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.
e^x
logistic differential equation - M = carrying capacity

6. y = cos²(3x)
e^x
chain rule
draw short segments representing slope at each point
relative minimum

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

8. Alternate definition of derivative
0/0 - 8/8 - 8*0 - 8 - 8 - 1^8 - 0° - 8°
product rule
general term = a1r^n - converges if -1 < r < 1
Limit as x approaches a of [f(x)-f(a)]/(x-a)

9. To draw a slope field - plug (x -y) coordinates into differential equation...
draw short segments representing slope at each point
? f(x) dx on interval a to b = F(b) - F(a)
? v (dx/dt)² + (dy/dt)² over interval from a to b
f(x) has a relative minimum

10. [(h1 - h2)/2]*base
speed
velocity is positive
Area of trapezoid
velocity is negative

11. Given velocity vectors dx/dt and dy/dt - find speed
v(dx/dt)² + (dy/dt)² not an integral!
Limit as x approaches a of [f(x)-f(a)]/(x-a)
logistic differential equation - M = carrying capacity
find first derivative - dy/dx = dy/dt / dx/dt - then find derivative of first derivative - then divide by dx/dt

12. y = sin?¹(x) - y' =

13. Alternating series tes
Slope of secant line between two points - use to estimate instantanous rate of change at a point.
lim as n approaches zero of general term = 0 and terms decrease - series converges
y' = -1/(1 + x²)
Limit as x approaches a of [f(x)-f(a)]/(x-a)

14. Particle is moving to the left/down
velocity is negative
y' = 1/(1 + x²)
Alternating series converges and general term converges with another test
p ? r² dx over interval a to b - where r = distance from curve to axis of revolution

15. Product Rule

16. Eatio test
relative maximum
lim as n approaches 8 of ratio of (n+1) term/nth term > 1 - series converges
polynomial with finite number of terms - largest exponent is 6 - find all derivatives up to the 6th derivative
y' = 1/v(1 - x²)

17. trapezoidal rule
use trapezoids to evaluate integrals (estimate area)
use to find indeterminate limits - find derivative of numerator and denominator separately then evaluate limit
derivative
logistic differential equation - M = carrying capacity

18. slope of vertical line
negative
undefined
corner - cusp - vertical tangent - discontinuity
logistic differential equation - M = carrying capacity

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

20. y = ln(x)/x² - state rule used to find derivative
? v (dx/dt)² + (dy/dt)² over interval from a to b
quotient rule
y' = cos(x)
uv' + vu'

21. area under a curve
derivative
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
positive

22. When f '(x) changes fro positive to negative - f(x) has a...
relative maximum
speed
1/(b-a) ? f(x) dx on interval a to b
two different types of functions are multiplied

23. Converges conditionally
? f(x) dx on interval a to b = F(b) - F(a)
positive
Limit as x approaches a of [f(x)-f(a)]/(x-a)
Alternating series converges and general term diverges with another test

24. Length of curve
use rectangles with left-endpoints to evaluate integral (estimate area)
f '(g(x)) g'(x)
? v (dx/dt)² + (dy/dt)² over interval from a to b
? v(1 + (dy/dx)²) dx over interval a to b

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

26. Use partial fractions to integrate when...
f(x) has a relative maximum
general term = a1r^n - converges if -1 < r < 1
y' = 1/v(1 - x²)
integrand is a rational function with a factorable denominator

27. area below x-axis is...
f '(g(x)) g'(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.
negative
y' = sec(x)tan(x)

28. Linearization
lim as n approaches zero of general term = 0 and terms decrease - series converges
substitution - parts - partial fractions
Limit as x approaches a of [f(x)-f(a)]/(x-a)
use tangent line to approximate values of the function

29. Taylor series
uv - ? v du
concave down
y' = -sin(x)
polynomial with infinite number of terms - includes general term

30. y = x cos(x) - state rule used to find derivative
if terms grow without bound - series diverges
product rule
concave up
p ? r² dx over interval a to b - where r = distance from curve to axis of revolution

31. Converges absolutely
polynomial with finite number of terms - largest exponent is 6 - find all derivatives up to the 6th derivative
logistic differential equation - M = carrying capacity
draw short segments representing slope at each point
Alternating series converges and general term converges with another test

32. When f '(x) is increasing - f(x) is...
v(dx/dt)² + (dy/dt)² not an integral!
concave up
? v (dx/dt)² + (dy/dt)² over interval from a to b
find first derivative - dy/dx = dy/dt / dx/dt - then find derivative of first derivative - then divide by dx/dt

33. Integral test
y' = 1/x
if integral converges - series converges
no limits - find antiderivative + C - use inital value to find C
? v (dx/dt)² + (dy/dt)² over interval from a to b

34. Formal definition of derivative
corner - cusp - vertical tangent - discontinuity
positive
no limits - find antiderivative + C - use inital value to find C
Limit as h approaches 0 of [f(a+h)-f(a)]/h

35. When f '(x) is positive - f(x) is...
increasing
velocity is negative
f '(g(x)) g'(x)
Alternating series converges and general term diverges with another test

36. P = M / (1 + Ae^(-Mkt))
lim as n approaches 8 of ratio of (n+1) term/nth term > 1 - series converges
Limit as x approaches a of [f(x)-f(a)]/(x-a)
y' = 1/v(1 - x²)
logistic growth equation

37. Given velocity vectors dx/dt and dy/dt - find total distance travelled
? A(x) dx over interval a to b - where A(x) is the area of the given cross-section in terms of x
substitution - parts - partial fractions
y' = sec²(x)
? v (dx/dt)² + (dy/dt)² over interval from a to b

38. ? u dv =
? A(x) dx over interval a to b - where A(x) is the area of the given cross-section in terms of x
if terms grow without bound - series diverges
y' = sec²(x)
uv - ? v du

39. y = a^x - y' = y' =
use tangent line to approximate values of the function
a^x ln(a)
product rule
has limits a & b - find antiderivative - F(b) - F(a)

40. indefinite integral
no limits - find antiderivative + C - use inital value to find C
1/(b-a) ? f(x) dx on interval a to b
Slope of secant line between two points - use to estimate instantanous rate of change at a point.
relative maximum

41. nth term test
uv' + vu'
logistic differential equation - M = carrying capacity
general term = 1/n^p - converges if p > 1
if terms grow without bound - series diverges

42. Area between two curves
increasing
critical points and endpoints
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

43. Chain Rule

44. When f '(x) is negative - f(x) is...
use tangent line to approximate values of the function
integrand is a rational function with a factorable denominator
f(x) has a relative maximum
decreasing

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

46. When f '(x) changes from negative to positive - f(x) has a...
relative minimum
y' = sec(x)tan(x)
polynomial with infinite number of terms - includes general term
? v(1 + (dy/dx)²) dx over interval a to b

47. right riemann sum
? f(x) dx on interval a to b = F(b) - F(a)
use rectangles with right-endpoints to evaluate integrals (estimate area)
? v (dx/dt)² + (dy/dt)² over interval from a to b
(uv'-vu')/v²

48. absolute value of velocity
speed
derivative
undefined
? f(x) dx integrate over interval a to b

49. Limit comparison test
(uv'-vu')/v²
f(x)
A function and it's derivative are in the integrand
if lim as n approaches 8 of ratio of comparison series/general term is positive and finite - then series behaves like comparison series

50. Indeterminate forms
logistic growth equation
use rectangles with right-endpoints to evaluate integrals (estimate area)
if terms grow without bound - series diverges
0/0 - 8/8 - 8*0 - 8 - 8 - 1^8 - 0° - 8°