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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 = ln(x)/x² - state rule used to find derivative
quotient rule
y' = 1/(x lna)
relative minimum
velocity is negative

2. slope of vertical line
logistic differential equation - M = carrying capacity
y' = 1/x
undefined
Alternating series converges and general term diverges with another test

3. Length of curve
? v(1 + (dy/dx)²) dx over interval a to b
polynomial with infinite number of terms - includes general term
Alternating series converges and general term converges with another test
velocity is negative

4. Particle is moving to the left/down
p ? r² dx over interval a to b - where r = distance from curve to axis of revolution
velocity is negative
Area of trapezoid
y' = 1/(x lna)

5. Intermediate Value Theorem
f(x) has a relative maximum
if terms grow without bound - series diverges
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 finite number of terms - largest exponent is 6 - find all derivatives up to the 6th derivative

6. Product Rule

7. Given velocity vectors dx/dt and dy/dt - find speed
1/2 ? R² - r² over interval from a to b - find a & b by setting equations equal - solve for theta.
critical points and endpoints
v(dx/dt)² + (dy/dt)² not an integral!
integrand is a rational function with a factorable denominator

8. trapezoidal rule
use trapezoids to evaluate integrals (estimate area)
y' = -sin(x)
v(dx/dt)² + (dy/dt)² not an integral!
f(x) has a relative minimum

9. slope of horizontal line
zero
two different types of functions are multiplied
positive
general term = a1r^n - converges if -1 < r < 1

10. Taylor series
(uv'-vu')/v²
y' = 1/x
y' = -csc²(x)
polynomial with infinite number of terms - includes general term

11. Limit comparison test
f(x) has a relative maximum
p ? r² dx over interval a to b - where r = distance from curve to axis of revolution
if integral converges - 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

12. Alternate definition of derivative
? f(x) - g(x) over interval a to b - where f(x) is top function and g(x) is bottom function
use tangent line to approximate values of the function
1/2 ? r² over interval from a to b - find a & b by setting r = 0 - solve for theta
Limit as x approaches a of [f(x)-f(a)]/(x-a)

13. indefinite integral
no limits - find antiderivative + C - use inital value to find C
(uv'-vu')/v²
use ratio test - set > 1 and solve absolute value equations - radius = center - endpoint
if lim as n approaches 8 of ratio of comparison series/general term is positive and finite - then series behaves like comparison series

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

15. To draw a slope field - plug (x -y) coordinates into differential equation...
? v(1 + (dy/dx)²) dx over interval a to b
draw short segments representing slope at each point
f(x)
y' = 1/x

16. Use partial fractions to integrate when...
f(x) has a relative maximum
y' = sec(x)tan(x)
integrand is a rational function with a factorable denominator
y' = -1/v(1 - x²)

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

18. use integration by parts when...
negative
if lim as n approaches 8 of ratio of comparison series/general term is positive and finite - then series behaves like comparison series
two different types of functions are multiplied
y' = 1/v(1 - x²)

19. Volume of solid of revolution - washer
if lim as n approaches 8 of ratio of comparison series/general term is positive and finite - then series behaves like comparison series
if integral converges - series converges
negative
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

20. Find interval of convergence
? v(1 + (dy/dx)²) dx over interval a to b
logistic differential equation - M = carrying capacity
use ratio test - set > 1 and solve absolute value equations - check endpoints
y' = -sin(x)

21. Area inside one polar curve and outside another polar curve
Slope of secant line between two points - use to estimate instantanous rate of change at a point.
separate variables - integrate + C - use initial condition to find C - solve for y
1/2 ? R² - r² over interval from a to b - find a & b by setting equations equal - solve for theta.
v(dx/dt)² + (dy/dt)² not an integral!

22. Area inside polar curve
corner - cusp - vertical tangent - discontinuity
? v (dx/dt)² + (dy/dt)² over interval from a to b
1/2 ? r² over interval from a to b - find a & b by setting r = 0 - solve for theta
? v(1 + (dy/dx)²) dx over interval a to b

23. When f '(x) changes from negative to positive - f(x) has a...
relative minimum
p ? r² dx over interval a to b - where r = distance from curve to axis of revolution
use rectangles with left-endpoints to evaluate integral (estimate area)
find first derivative - dy/dx = dy/dt / dx/dt - then find derivative of first derivative - then divide by dx/dt

24. Alternating series tes
lim as n approaches zero of general term = 0 and terms decrease - series converges
decreasing
uv' + vu'
substitution - parts - partial fractions

25. Converges absolutely
has limits a & b - find antiderivative - F(b) - F(a)
f(x) has a relative minimum
Alternating series converges and general term converges with another test
integrand is a rational function with a factorable denominator

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

27. area under a curve
relative minimum
derivative
? f(x) dx integrate over interval a to b
Alternating series converges and general term converges with another test

28. When f '(x) is positive - f(x) is...
increasing
Slope of secant line between two points - use to estimate instantanous rate of change at a point.
use tangent line to approximate values of the function
lim as n approaches zero of general term = 0 and terms decrease - series converges

29. Formal definition of derivative
A function and it's derivative are in the integrand
Limit as h approaches 0 of [f(a+h)-f(a)]/h
two different types of functions are multiplied
if integral converges - series converges

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

31. area above x-axis is...
two different types of functions are multiplied
Slope of tangent line at a point - value of derivative at a point
positive
y' = sec²(x)

32. definite integral
has limits a & b - find antiderivative - F(b) - F(a)
? abs[v(t)] over interval a to b
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.
zero

33. y = cos?¹(x) - y' =

34. y = tan?¹(x) - y' =

35. When is a function not differentiable
corner - cusp - vertical tangent - discontinuity
use ratio test - set > 1 and solve absolute value equations - radius = center - endpoint
f(x) has a relative maximum
Slope of secant line between two points - use to estimate instantanous rate of change at a point.

36. Area between two curves
use to find indeterminate limits - find derivative of numerator and denominator separately then evaluate limit
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
y' = -sin(x)

37. Volume of solid with base in the plane and given cross-section
p ? r² dx over interval a to b - where r = distance from curve to axis of revolution
y' = 1/(1 + x²)
? A(x) dx over interval a to b - where A(x) is the area of the given cross-section in terms of x
f(x) has a relative minimum

38. Eatio test
undefined
lim as n approaches 8 of ratio of (n+1) term/nth term > 1 - series converges
derivative
Slope of secant line between two points - use to estimate instantanous rate of change at a point.

39. rate
derivative
a^x ln(a)
y' = -1/v(1 - x²)
1/(b-a) ? f(x) dx on interval a to b

40. When f '(x) is negative - f(x) is...
(uv'-vu')/v²
y' = 1/v(1 - x²)
decreasing
relative minimum

41. To find particular solution to differential equation - dy/dx = x/y...
derivative
separate variables - integrate + C - use initial condition to find C - solve for y
? v(1 + (dy/dx)²) dx over interval a to b
uv - ? v du

42. Find radius of convergence
use ratio test - set > 1 and solve absolute value equations - radius = center - endpoint
use tangent line to approximate values of the function
? v(t) over interval a to b
Alternating series converges and general term diverges with another test

43. y = sin?¹(x) - y' =

44. When f '(x) is decreasing - f(x) is...
concave down
integrand is a rational function with a factorable denominator
polynomial with infinite number of terms - includes general term
logistic differential equation - M = carrying capacity

45. y = cos²(3x)
use tangent line to approximate values of the function
uv' + vu'
chain rule
use to find indeterminate limits - find derivative of numerator and denominator separately then evaluate limit

46. Particle is moving to the right/up
velocity is positive
logistic growth equation
y' = sec²(x)
critical points and endpoints

47. Linearization
integrand is a rational function with a factorable denominator
use trapezoids to evaluate integrals (estimate area)
uv' + vu'
use tangent line to approximate values of the function

48. 6th degree Taylor Polynomial
use ratio test - set > 1 and solve absolute value equations - check endpoints
A function and it's derivative are in the integrand
logistic differential equation - M = carrying capacity
polynomial with finite number of terms - largest exponent is 6 - find all derivatives up to the 6th derivative

49. Indeterminate forms
quotient rule
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.
0/0 - 8/8 - 8*0 - 8 - 8 - 1^8 - 0° - 8°

50. use substitution to integrate when

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