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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. Use partial fractions to integrate when...
a^x ln(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.
y' = 1/(x lna)
integrand is a rational function with a factorable denominator

2. Alternate definition of derivative
Limit as x approaches a of [f(x)-f(a)]/(x-a)
decreasing
y' = cos(x)
integrand is a rational function with a factorable denominator

3. p-series test
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 rectangles with left-endpoints to evaluate integral (estimate area)
general term = 1/n^p - converges if p > 1
has limits a & b - find antiderivative - F(b) - F(a)

4. Given velocity vectors dx/dt and dy/dt - find speed
? v(t) over interval a to b
use rectangles with right-endpoints to evaluate integrals (estimate area)
v(dx/dt)² + (dy/dt)² not an integral!
undefined

5. Given v(t) 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
zero
general term = 1/n^p - converges if p > 1
? abs[v(t)] over interval a to b

6. Quotient Rule

7. slope of horizontal line
1/2 ? r² over interval from a to b - find a & b by setting r = 0 - solve for theta
zero
substitution - parts - partial fractions
? v(t) over interval a to b

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

9. Volume of solid of revolution - washer
use tangent line to approximate values of the function
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
product rule

10. Length of parametric curve
logistic differential equation - M = carrying capacity
? v (dx/dt)² + (dy/dt)² over interval from a to b
uv' + vu'
a^x ln(a)

11. left riemann sum
? v(t) over interval a to b
use rectangles with left-endpoints to evaluate integral (estimate area)
derivative
zero

12. Particle is moving to the right/up
velocity is positive
y' = -1/(1 + x²)
y' = -sin(x)
y' = 1/(1 + x²)

13. Fundamental Theorem of Calculus
? f(x) dx on interval a to b = F(b) - F(a)
uv - ? v du
? v (dx/dt)² + (dy/dt)² over interval from a to b
integrand is a rational function with a factorable denominator

14. average value of f(x)
1/(b-a) ? f(x) dx on interval a to b
integrand is a rational function with a factorable denominator
lim as n approaches 8 of ratio of (n+1) term/nth term > 1 - series converges
y' = -csc²(x)

15. When is a function not differentiable
corner - cusp - vertical tangent - discontinuity
1/(b-a) ? f(x) dx on interval a to b
lim as n approaches zero of general term = 0 and terms decrease - series converges
y' = cos(x)

16. When f '(x) is decreasing - f(x) is...
point of inflection
a^x ln(a)
has limits a & b - find antiderivative - F(b) - F(a)
concave down

17. slope of vertical line
if terms grow without bound - series diverges
1/2 ? R² - r² over interval from a to b - find a & b by setting equations equal - solve for theta.
f(x) has a relative maximum
undefined

18. When f '(x) is increasing - f(x) is...
concave up
f(x)
y' = sec(x)tan(x)
f(x) has a relative minimum

19. Area inside polar curve
y' = 1/v(1 - x²)
1/2 ? r² over interval from a to b - find a & b by setting r = 0 - solve for theta
substitution - parts - partial fractions
increasing

20. right riemann sum
0/0 - 8/8 - 8*0 - 8 - 8 - 1^8 - 0° - 8°
use rectangles with right-endpoints to evaluate integrals (estimate area)
? v(t) over interval a to b
y' = cos(x)

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

22. definite integral
Limit as h approaches 0 of [f(a+h)-f(a)]/h
if terms grow without bound - series diverges
has limits a & b - find antiderivative - F(b) - F(a)
Limit as x approaches a of [f(x)-f(a)]/(x-a)

23. P = M / (1 + Ae^(-Mkt))
logistic growth equation
if lim as n approaches 8 of ratio of comparison series/general term is positive and finite - then series behaves like comparison series
zero
a^x ln(a)

24. indefinite integral
undefined
Slope of tangent line at a point - value of derivative at a point
use trapezoids to evaluate integrals (estimate area)
no limits - find antiderivative + C - use inital value to find C

25. y = cos²(3x)
Limit as x approaches a of [f(x)-f(a)]/(x-a)
velocity is positive
chain rule
lim as n approaches 8 of ratio of (n+1) term/nth term > 1 - series converges

26. To find absolute maximum on closed interval [a - b] - you must consider...
negative
concave down
Limit as h approaches 0 of [f(a+h)-f(a)]/h
critical points and endpoints

27. area below x-axis is...
positive
point of inflection
y' = 1/v(1 - x²)
negative

28. [(h1 - h2)/2]*base
if integral converges - series converges
y' = -1/v(1 - x²)
Area of trapezoid
general term = a1r^n - converges if -1 < r < 1

29. Converges absolutely
use rectangles with right-endpoints to evaluate integrals (estimate area)
chain rule
Alternating series converges and general term converges with another test
f(x) has a relative maximum

30. If f '(x) = 0 and f'(x) < 0 -...
f(x) has a relative maximum
Alternating series converges and general term diverges with another test
(uv'-vu')/v²
integrand is a rational function with a factorable denominator

31. When f '(x) changes from increasing to decreasing or decreasing to increasing - f(x) has a...
use rectangles with right-endpoints to evaluate integrals (estimate area)
if lim as n approaches 8 of ratio of comparison series/general term is positive and finite - then series behaves like comparison series
use trapezoids to evaluate integrals (estimate area)
point of inflection

32. Integral test
separate variables - integrate + C - use initial condition to find C - solve for y
if integral converges - series converges
? f(x) dx integrate over interval a to b
negative

33. ? u dv =
v(dx/dt)² + (dy/dt)² not an integral!
uv - ? v du
Slope of tangent line at a point - value of derivative at a point
speed

34. Average Rate of Change
Slope of secant line between two points - use to estimate instantanous rate of change at a point.
f(x) has a relative minimum
logistic differential equation - M = carrying capacity
corner - cusp - vertical tangent - discontinuity

35. y = tan?¹(x) - y' =

36. To draw a slope field - plug (x -y) coordinates into differential equation...
draw short segments representing slope at each point
? A(x) dx over interval a to b - where A(x) is the area of the given cross-section in terms of x
point of inflection
Area of trapezoid

37. Converges conditionally
no limits - find antiderivative + C - use inital value to find C
quotient rule
? abs[v(t)] over interval a to b
Alternating series converges and general term diverges with another test

38. Geometric series test
if lim as n approaches 8 of ratio of comparison series/general term is positive and finite - then series behaves like comparison series
general term = a1r^n - converges if -1 < r < 1
polynomial with finite number of terms - largest exponent is 6 - find all derivatives up to the 6th derivative
1/2 ? r² over interval from a to b - find a & b by setting r = 0 - solve for theta

39. y = ln(x)/x² - state rule used to find derivative
uv - ? v du
quotient rule
Slope of tangent line at a point - value of derivative at a point
y' = 1/(1 + x²)

40. Intermediate Value Theorem
general term = a1r^n - converges if -1 < r < 1
logistic growth equation
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.
relative minimum

41. area above x-axis is...
positive
y' = 1/(x lna)
concave up
uv' + vu'

42. When f '(x) is negative - f(x) is...
Slope of tangent line at a point - value of derivative at a point
product rule
velocity is positive
decreasing

43. Indeterminate forms
uv' + vu'
? v(t) over interval a to b
0/0 - 8/8 - 8*0 - 8 - 8 - 1^8 - 0° - 8°
Alternating series converges and general term diverges with another test

44. use substitution to integrate when

45. When f '(x) is positive - f(x) is...
(uv'-vu')/v²
draw short segments representing slope at each point
use rectangles with right-endpoints to evaluate integrals (estimate area)
increasing

46. nth term test
use to find indeterminate limits - find derivative of numerator and denominator separately then evaluate limit
y' = -1/(1 + x²)
if terms grow without bound - series diverges
? f(x) dx integrate over interval a to b

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

48. Formal definition of derivative
y' = -sin(x)
if terms grow without bound - series diverges
Limit as h approaches 0 of [f(a+h)-f(a)]/h
negative

49. Instantenous Rate of Change
two different types of functions are multiplied
Limit as h approaches 0 of [f(a+h)-f(a)]/h
Slope of tangent line at a point - value of derivative at a point
velocity is positive

50. Area inside one polar curve and outside another polar curve
chain rule
f(x) has a relative minimum
1/2 ? R² - r² over interval from a to b - find a & b by setting equations equal - solve for theta.
y' = sec(x)tan(x)