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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. y = e^x - y' = y' =
e^x
A function and it's derivative are in the integrand
derivative
? v (dx/dt)² + (dy/dt)² over interval from a to b

2. right riemann sum
? v (dx/dt)² + (dy/dt)² over interval from a to b
has limits a & b - find antiderivative - F(b) - F(a)
use rectangles with right-endpoints to evaluate integrals (estimate area)
? abs[v(t)] over interval a to b

3. To find particular solution to differential equation - dy/dx = x/y...
e^x
Limit as h approaches 0 of [f(a+h)-f(a)]/h
separate variables - integrate + C - use initial condition to find C - solve for y
negative

4. Given velocity vectors dx/dt and dy/dt - find speed
if integral converges - series converges
y' = 1/v(1 - x²)
y' = sec²(x)
v(dx/dt)² + (dy/dt)² not an integral!

5. If f '(x) = 0 and f'(x) < 0 -...
? v (dx/dt)² + (dy/dt)² over interval from a to b
corner - cusp - vertical tangent - discontinuity
f(x) has a relative maximum
relative maximum

6. area below x-axis is...
e^x
negative
y' = -csc(x)cot(x)
lim as n approaches zero of general term = 0 and terms decrease - series converges

7. Length of parametric curve
? v (dx/dt)² + (dy/dt)² over interval from a to b
quotient rule
use tangent line to approximate values of the function
if lim as n approaches 8 of ratio of comparison series/general term is positive and finite - then series behaves like comparison series

8. [(h1 - h2)/2]*base
use tangent line to approximate values of the function
Alternating series converges and general term converges with another test
Area of trapezoid
product rule

9. When f '(x) changes fro positive to negative - f(x) has a...
? 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
relative maximum
integrand is a rational function with a factorable denominator

10. When f '(x) is increasing - f(x) is...
f(x) has a relative maximum
concave up
if lim as n approaches 8 of ratio of comparison series/general term is positive and finite - then series behaves like comparison series
increasing

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

12. Integral test
if integral converges - series converges
v(dx/dt)² + (dy/dt)² not an integral!
y' = sec(x)tan(x)
0/0 - 8/8 - 8*0 - 8 - 8 - 1^8 - 0° - 8°

13. Second derivative of parametrically defined curve
find first derivative - dy/dx = dy/dt / dx/dt - then find derivative of first derivative - then divide by dx/dt
if lim as n approaches 8 of ratio of comparison series/general term is positive and finite - then series behaves like comparison series
v(dx/dt)² + (dy/dt)² not an integral!
if terms grow without bound - series diverges

14. When f '(x) changes from negative to positive - f(x) has a...
corner - cusp - vertical tangent - discontinuity
relative minimum
derivative
logistic growth equation

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

16. Given v(t) find total distance travelled
a^x ln(a)
y' = -1/v(1 - x²)
y' = -sin(x)
? abs[v(t)] over interval a to b

17. Volume of solid of revolution - washer
zero
? f(x) - g(x) over interval a to b - where f(x) is top function and g(x) is bottom function
y' = 1/(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

18. definite integral
integrand is a rational function with a factorable denominator
? v(t) over interval a to b
has limits a & b - find antiderivative - F(b) - F(a)
positive

19. Area inside one polar curve and outside another polar curve
has limits a & b - find antiderivative - F(b) - F(a)
concave up
1/2 ? R² - r² over interval from a to b - find a & b by setting equations equal - solve for theta.
y' = 1/(1 + x²)

20. If f '(x) = 0 and f'(x) > 0 -...
Slope of secant line between two points - use to estimate instantanous rate of change at a point.
f(x) has a relative minimum
0/0 - 8/8 - 8*0 - 8 - 8 - 1^8 - 0° - 8°
if terms grow without bound - series diverges

21. When f '(x) is decreasing - f(x) is...
concave down
uv - ? v du
has limits a & b - find antiderivative - F(b) - F(a)
logistic growth equation

22. Length of curve
? v(1 + (dy/dx)²) dx over interval a to b
substitution - parts - partial fractions
use trapezoids to evaluate integrals (estimate area)
find first derivative - dy/dx = dy/dt / dx/dt - then find derivative of first derivative - then divide by dx/dt

23. Taylor series
use rectangles with left-endpoints to evaluate integral (estimate area)
zero
polynomial with infinite number of terms - includes general term
? v(1 + (dy/dx)²) dx over interval a to b

24. 6th degree Taylor Polynomial
polynomial with finite number of terms - largest exponent is 6 - find all derivatives up to the 6th derivative
has limits a & b - find antiderivative - F(b) - F(a)
uv - ? v du
uv' + vu'

25. y = sin?¹(x) - y' =

26. Particle is moving to the left/down
critical points and endpoints
general term = a1r^n - converges if -1 < r < 1
velocity is negative
speed

27. If g(x) = ? f(t) dt on interval 2 to x - then g'(x) =...
relative minimum
f '(g(x)) g'(x)
f(x)
? v (dx/dt)² + (dy/dt)² over interval from a to b

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

29. Average Rate of Change
find first derivative - dy/dx = dy/dt / dx/dt - then find derivative of first derivative - then divide by dx/dt
Slope of secant line between two points - use to estimate instantanous rate of change at a point.
p ? r² dx over interval a to b - where r = distance from curve to axis of revolution
zero

30. Alternating series tes
relative minimum
lim as n approaches zero of general term = 0 and terms decrease - series converges
decreasing
critical points and endpoints

31. Converges conditionally
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
Alternating series converges and general term diverges with another test
y' = -1/v(1 - x²)

32. Quotient Rule

33. Intermediate Value Theorem
chain rule
find first derivative - dy/dx = dy/dt / dx/dt - then find derivative of first derivative - then divide by dx/dt
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.

34. When f '(x) is positive - f(x) is...
integrand is a rational function with a factorable denominator
y' = -csc(x)cot(x)
quotient rule
increasing

35. ? u dv =
y' = -csc(x)cot(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.
uv - ? v du
if terms grow without bound - series diverges

36. y = a^x - y' = y' =
lim as n approaches zero of general term = 0 and terms decrease - series converges
use ratio test - set > 1 and solve absolute value equations - check endpoints
a^x ln(a)
speed

37. Area between two curves
0/0 - 8/8 - 8*0 - 8 - 8 - 1^8 - 0° - 8°
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
v(dx/dt)² + (dy/dt)² not an integral!

38. Eatio test
speed
lim as n approaches 8 of ratio of (n+1) term/nth term > 1 - series converges
1/(b-a) ? f(x) dx on interval a to b
? v(t) over interval a to b

39. Fundamental Theorem of Calculus
concave up
if terms grow without bound - series diverges
? 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

40. nth term test
if terms grow without bound - series diverges
increasing
uv' + vu'
? A(x) dx over interval a to b - where A(x) is the area of the given cross-section in terms of x

41. P = M / (1 + Ae^(-Mkt))
1/2 ? r² over interval from a to b - find a & b by setting r = 0 - solve for theta
y' = 1/(1 + x²)
logistic growth equation
Slope of secant line between two points - use to estimate instantanous rate of change at a point.

42. Volume of solid with base in the plane and given cross-section
polynomial with infinite number of terms - includes general term
Area of trapezoid
? A(x) dx over interval a to b - where A(x) is the area of the given cross-section in terms of x
? v (dx/dt)² + (dy/dt)² over interval from a to b

43. methods of integration
substitution - parts - partial fractions
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.
corner - cusp - vertical tangent - discontinuity
general term = 1/n^p - converges if p > 1

44. slope of vertical line
negative
undefined
y' = 1/(1 + x²)
use ratio test - set > 1 and solve absolute value equations - radius = center - endpoint

45. Product Rule

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

47. trapezoidal rule
(uv'-vu')/v²
use rectangles with left-endpoints to evaluate integral (estimate area)
use trapezoids to evaluate integrals (estimate area)
if terms grow without bound - series diverges

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

49. Given v(t) find displacement
? v(t) over interval a to b
positive
use ratio test - set > 1 and solve absolute value equations - radius = center - endpoint
critical points and endpoints

50. Converges absolutely
0/0 - 8/8 - 8*0 - 8 - 8 - 1^8 - 0° - 8°
Alternating series converges and general term converges with another test
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