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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. Integral test
uv - ? v du
Slope of secant line between two points - use to estimate instantanous rate of change at a point.
find first derivative - dy/dx = dy/dt / dx/dt - then find derivative of first derivative - then divide by dx/dt
if integral converges - series converges

2. When is a function not differentiable
y' = sec(x)tan(x)
corner - cusp - vertical tangent - discontinuity
decreasing
y' = -sin(x)

3. Instantenous Rate of Change
substitution - parts - partial fractions
? f(x) dx on interval a to b = F(b) - F(a)
Slope of tangent line at a point - value of derivative at a point
polynomial with finite number of terms - largest exponent is 6 - find all derivatives up to the 6th derivative

4. use integration by parts when...
two different types of functions are multiplied
negative
draw short segments representing slope at each point
0/0 - 8/8 - 8*0 - 8 - 8 - 1^8 - 0° - 8°

5. Given v(t) find displacement
point of inflection
? v(t) over interval a to b
? f(x) dx integrate over interval a to b
1/2 ? R² - r² over interval from a to b - find a & b by setting equations equal - solve for theta.

6. To draw a slope field - plug (x -y) coordinates into differential equation...
(uv'-vu')/v²
draw short segments representing slope at each point
f(x) has a relative maximum
Alternating series converges and general term diverges with another test

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

8. If f '(x) = 0 and f'(x) > 0 -...
separate variables - integrate + C - use initial condition to find C - solve for y
f(x) has a relative maximum
f(x) has a relative minimum
y' = -sin(x)

9. When f '(x) changes from increasing to decreasing or decreasing to increasing - f(x) has a...
critical points and endpoints
? f(x) dx on interval a to b = F(b) - F(a)
point of inflection
general term = 1/n^p - converges if p > 1

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

11. ? u dv =
uv - ? v du
y' = -1/(1 + x²)
use ratio test - set > 1 and solve absolute value equations - check endpoints
general term = 1/n^p - converges if p > 1

12. Volume of solid of revolution - washer
f(x) has a relative maximum
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
? 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)

13. Area between two curves
e^x
Area of trapezoid
uv - ? v du
? f(x) - g(x) over interval a to b - where f(x) is top function and g(x) is bottom function

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

15. [(h1 - h2)/2]*base
a^x ln(a)
y' = sec(x)tan(x)
speed
Area of trapezoid

16. Converges conditionally
draw short segments representing slope at each point
Alternating series converges and general term diverges with another test
Alternating series converges and general term converges with another test
two different types of functions are multiplied

17. y = cos²(3x)
? f(x) dx on interval a to b = F(b) - F(a)
a^x ln(a)
p ? r² dx over interval a to b - where r = distance from curve to axis of revolution
chain rule

18. p-series test
Alternating series converges and general term converges with another test
general term = 1/n^p - converges if p > 1
velocity is positive
y' = -1/(1 + x²)

19. Area inside polar curve
find first derivative - dy/dx = dy/dt / dx/dt - then find derivative of first derivative - then divide by dx/dt
lim as n approaches zero of general term = 0 and terms decrease - series converges
concave up
1/2 ? r² over interval from a to b - find a & b by setting r = 0 - solve for theta

20. mean value theorem

21. indefinite integral
y' = -sin(x)
lim as n approaches zero of general term = 0 and terms decrease - series converges
Limit as h approaches 0 of [f(a+h)-f(a)]/h
no limits - find antiderivative + C - use inital value to find C

22. Fundamental Theorem of Calculus
? f(x) dx on interval a to b = F(b) - F(a)
y' = 1/v(1 - x²)
chain rule
lim as n approaches 8 of ratio of (n+1) term/nth term > 1 - series converges

23. y = e^x - y' = y' =
e^x
corner - cusp - vertical tangent - discontinuity
lim as n approaches zero of general term = 0 and terms decrease - series converges
y' = -1/(1 + x²)

24. Particle is moving to the left/down
A function and it's derivative are in the integrand
velocity is negative
chain rule
y' = sec(x)tan(x)

25. y = x cos(x) - state rule used to find derivative
lim as n approaches 8 of ratio of (n+1) term/nth term > 1 - series converges
speed
? v(1 + (dy/dx)²) dx over interval a to b
product rule

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

27. Area inside one polar curve and outside another polar curve
? f(x) dx integrate over interval a to b
point of inflection
Slope of tangent line at a point - value of derivative at a point
1/2 ? R² - r² over interval from a to b - find a & b by setting equations equal - solve for theta.

28. Average Rate of Change
derivative
Slope of secant line between two points - use to estimate instantanous rate of change at a point.
find first derivative - dy/dx = dy/dt / dx/dt - then find derivative of first derivative - then divide by dx/dt
logistic growth equation

29. Given v(t) find total distance travelled
f(x) has a relative maximum
f(x)
y' = -1/v(1 - x²)
? abs[v(t)] over interval a to b

30. When f '(x) changes fro positive to negative - f(x) has a...
use ratio test - set > 1 and solve absolute value equations - radius = center - endpoint
relative maximum
y' = -csc(x)cot(x)
use ratio test - set > 1 and solve absolute value equations - check endpoints

31. 6th degree Taylor Polynomial
logistic differential equation - M = carrying capacity
1/(b-a) ? f(x) dx on interval a to b
? abs[v(t)] over interval a to b
polynomial with finite number of terms - largest exponent is 6 - find all derivatives up to the 6th derivative

32. y = sin?¹(x) - y' =

33. Find interval of convergence
use ratio test - set > 1 and solve absolute value equations - check endpoints
chain rule
negative
Alternating series converges and general term converges with another test

34. Linearization
use tangent line to approximate values of the function
? 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
A function and it's derivative are in the integrand

35. Alternating series tes
Alternating series converges and general term converges with another test
lim as n approaches zero of general term = 0 and terms decrease - series converges
general term = 1/n^p - converges if p > 1
a^x ln(a)

36. definite integral
concave up
has limits a & b - find antiderivative - F(b) - F(a)
no limits - find antiderivative + C - use inital value to find C
Alternating series converges and general term diverges with another test

37. To find particular solution to differential equation - dy/dx = x/y...
A function and it's derivative are in the integrand
Area of trapezoid
separate variables - integrate + C - use initial condition to find C - solve for y
relative minimum

38. When f '(x) is increasing - f(x) is...
use tangent line to approximate values of the function
negative
polynomial with finite number of terms - largest exponent is 6 - find all derivatives up to the 6th derivative
concave up

39. absolute value of velocity
integrand is a rational function with a factorable denominator
undefined
speed
? v (dx/dt)² + (dy/dt)² over interval from a to b

40. Length of parametric curve
f '(g(x)) g'(x)
lim as n approaches zero of general term = 0 and terms decrease - series converges
? v (dx/dt)² + (dy/dt)² over interval from a to b
chain rule

41. When f '(x) changes from negative to positive - f(x) has a...
relative minimum
decreasing
1/2 ? R² - r² over interval from a to b - find a & b by setting equations equal - solve for theta.
1/2 ? r² over interval from a to b - find a & b by setting r = 0 - solve for theta

42. nth term test
if terms grow without bound - series diverges
? abs[v(t)] over interval a to b
Alternating series converges and general term diverges with another test
increasing

43. Given velocity vectors dx/dt and dy/dt - find speed
v(dx/dt)² + (dy/dt)² not an integral!
relative maximum
polynomial with infinite number of terms - includes general term
(uv'-vu')/v²

44. use substitution to integrate when

45. Converges absolutely
f(x)
Alternating series converges and general term converges with another test
find first derivative - dy/dx = dy/dt / dx/dt - then find derivative of first derivative - then divide by dx/dt
chain rule

46. slope of horizontal line
if terms grow without bound - series diverges
zero
critical points and endpoints
Limit as x approaches a of [f(x)-f(a)]/(x-a)

47. Chain Rule

48. Formal definition of derivative
y' = cos(x)
no limits - find antiderivative + C - use inital value to find C
Limit as h approaches 0 of [f(a+h)-f(a)]/h
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.

49. Intermediate Value Theorem
critical points and endpoints
integrand is a rational function with a factorable denominator
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.

50. Use partial fractions to integrate when...
point of inflection
integrand is a rational function with a factorable denominator
y' = 1/x
general term = 1/n^p - converges if p > 1