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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. ? u dv =
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
no limits - find antiderivative + C - use inital value to find C
uv - ? v du

2. When f '(x) is negative - f(x) is...
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.
integrand is a rational function with a factorable denominator
decreasing
1/2 ? r² over interval from a to b - find a & b by setting r = 0 - solve for theta

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

4. Use partial fractions to integrate when...
use tangent line to approximate values of the function
y' = -1/(1 + x²)
speed
integrand is a rational function with a factorable denominator

5. When f '(x) changes from negative to positive - f(x) has a...
relative minimum
use rectangles with right-endpoints to evaluate integrals (estimate area)
? v(1 + (dy/dx)²) dx over interval a to b
use to find indeterminate limits - find derivative of numerator and denominator separately then evaluate limit

6. absolute value of velocity
y' = cos(x)
1/2 ? R² - r² over interval from a to b - find a & b by setting equations equal - solve for theta.
speed
v(dx/dt)² + (dy/dt)² not an integral!

7. P = M / (1 + Ae^(-Mkt))
logistic growth equation
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
y' = -csc(x)cot(x)
f(x) has a relative minimum

8. average value of f(x)
Limit as h approaches 0 of [f(a+h)-f(a)]/h
? v (dx/dt)² + (dy/dt)² over interval from a to b
(uv'-vu')/v²
1/(b-a) ? f(x) dx on interval a to b

9. Geometric series test
f '(g(x)) g'(x)
general term = a1r^n - converges if -1 < r < 1
use rectangles with left-endpoints to evaluate integral (estimate area)
lim as n approaches 8 of ratio of (n+1) term/nth term > 1 - series converges

10. Given v(t) find total distance travelled
f(x) has a relative maximum
? abs[v(t)] over interval a to b
corner - cusp - vertical tangent - discontinuity
1/2 ? R² - r² over interval from a to b - find a & b by setting equations equal - solve for theta.

11. Given velocity vectors dx/dt and dy/dt - find speed
v(dx/dt)² + (dy/dt)² not an integral!
y' = 1/(1 + x²)
? f(x) dx integrate over interval a to b
? f(x) - g(x) over interval a to b - where f(x) is top function and g(x) is bottom function

12. Converges absolutely
f(x)
two different types of functions are multiplied
quotient rule
Alternating series converges and general term converges with another test

13. Integral test
if integral converges - series converges
critical points and endpoints
f(x)
f '(g(x)) g'(x)

14. slope of vertical line
corner - cusp - vertical tangent - discontinuity
y' = -csc(x)cot(x)
undefined
negative

15. 6th degree Taylor Polynomial
polynomial with finite number of terms - largest exponent is 6 - find all derivatives up to the 6th derivative
chain rule
y' = -1/v(1 - x²)
positive

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

17. y = sin?¹(x) - y' =

18. definite integral
quotient rule
concave up
has limits a & b - find antiderivative - F(b) - F(a)
undefined

19. area under a curve
derivative
? f(x) dx integrate over interval a to b
Slope of tangent line at a point - value of derivative at a point
y' = 1/v(1 - x²)

20. 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
Limit as h approaches 0 of [f(a+h)-f(a)]/h
has limits a & b - find antiderivative - F(b) - F(a)
1/2 ? r² over interval from a to b - find a & b by setting r = 0 - solve for theta

21. If f '(x) = 0 and f'(x) > 0 -...
v(dx/dt)² + (dy/dt)² not an integral!
f(x) has a relative minimum
negative
y' = 1/(x lna)

22. When f '(x) is increasing - f(x) is...
use tangent line to approximate values of the function
y' = -1/v(1 - x²)
? f(x) dx on interval a to b = F(b) - F(a)
concave up

23. dP/dt = kP(M - P)
Alternating series converges and general term diverges with another test
lim as n approaches 8 of ratio of (n+1) term/nth term > 1 - series converges
e^x
logistic differential equation - M = carrying capacity

24. Volume of solid with base in the plane and given cross-section
? A(x) dx over interval a to b - where A(x) is the area of the given cross-section in terms of x
polynomial with infinite number of terms - includes general term
p ? r² dx over interval a to b - where r = distance from curve to axis of revolution
Alternating series converges and general term converges with another test

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

26. When f '(x) changes fro positive to negative - f(x) has a...
y' = -csc²(x)
if lim as n approaches 8 of ratio of comparison series/general term is positive and finite - then series behaves like comparison series
relative maximum
integrand is a rational function with a factorable denominator

27. To find particular solution to differential equation - dy/dx = x/y...
lim as n approaches zero of general term = 0 and terms decrease - series converges
separate variables - integrate + C - use initial condition to find C - solve for y
? f(x) - g(x) over interval a to b - where f(x) is top function and g(x) is bottom function
speed

28. right riemann sum
use rectangles with right-endpoints to evaluate integrals (estimate area)
use ratio test - set > 1 and solve absolute value equations - radius = center - endpoint
a^x ln(a)
f '(g(x)) g'(x)

29. Alternate definition of derivative
Limit as x approaches a of [f(x)-f(a)]/(x-a)
point of inflection
f(x) has a relative maximum
general term = a1r^n - converges if -1 < r < 1

30. Instantenous Rate of Change
derivative
uv' + vu'
Slope of secant line between two points - use to estimate instantanous rate of change at a point.
Slope of tangent line at a point - value of derivative at a point

31. When f '(x) is decreasing - f(x) is...
relative maximum
1/2 ? r² over interval from a to b - find a & b by setting r = 0 - solve for theta
product rule
concave down

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

33. indefinite integral
y' = 1/v(1 - x²)
no limits - find antiderivative + C - use inital value to find C
relative maximum
f(x) has a relative maximum

34. trapezoidal rule
? v (dx/dt)² + (dy/dt)² over interval from a to b
point of inflection
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)
use trapezoids to evaluate integrals (estimate area)

35. Eatio test
draw short segments representing slope at each point
? v (dx/dt)² + (dy/dt)² over interval from a to b
f '(g(x)) g'(x)
lim as n approaches 8 of ratio of (n+1) term/nth term > 1 - series converges

36. Given velocity vectors dx/dt and dy/dt - find total distance travelled
A function and it's derivative are in the integrand
0/0 - 8/8 - 8*0 - 8 - 8 - 1^8 - 0° - 8°
? f(x) dx integrate over interval a to b
? v (dx/dt)² + (dy/dt)² over interval from a to b

37. y = a^x - y' = y' =
a^x ln(a)
if lim as n approaches 8 of ratio of comparison series/general term is positive and finite - then series behaves like comparison series
concave up
Slope of tangent line at a point - value of derivative at a point

38. left riemann sum
relative maximum
if lim as n approaches 8 of ratio of comparison series/general term is positive and finite - then series behaves like comparison series
use rectangles with left-endpoints to evaluate integral (estimate area)
concave up

39. Taylor series
draw short segments representing slope at each point
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
polynomial with infinite number of terms - includes general term
velocity is positive

40. Quotient Rule

41. methods of integration
lim as n approaches zero of general term = 0 and terms decrease - series converges
substitution - parts - partial fractions
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) dx on interval a to b = F(b) - F(a)

42. slope of horizontal line
A function and it's derivative are in the integrand
use to find indeterminate limits - find derivative of numerator and denominator separately then evaluate limit
zero
substitution - parts - partial fractions

43. Area inside one polar curve and outside another polar curve
if lim as n approaches 8 of ratio of comparison series/general term is positive and finite - then series behaves like comparison series
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.
point of inflection

44. Intermediate Value Theorem
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.
draw short segments representing slope at each point
Alternating series converges and general term diverges with another test
general term = a1r^n - converges if -1 < r < 1

45. Indeterminate forms
y' = cos(x)
0/0 - 8/8 - 8*0 - 8 - 8 - 1^8 - 0° - 8°
general term = 1/n^p - converges if p > 1
? f(x) dx integrate over interval a to b

46. y = x cos(x) - state rule used to find derivative
if terms grow without bound - series diverges
product rule
use ratio test - set > 1 and solve absolute value equations - check endpoints
logistic growth equation

47. If f '(x) = 0 and f'(x) < 0 -...
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.
f(x) has a relative maximum
no limits - find antiderivative + C - use inital value to find C
zero

48. y = ln(x)/x² - state rule used to find derivative
1/(b-a) ? f(x) dx on interval a to b
p ? r² dx over interval a to b - where r = distance from curve to axis of revolution
y' = 1/x
quotient rule

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

50. Area inside polar curve
corner - cusp - vertical tangent - discontinuity
1/2 ? r² over interval from a to b - find a & b by setting r = 0 - solve for theta
? f(x) - g(x) over interval a to b - where f(x) is top function and g(x) is bottom function
y' = -csc²(x)