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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. When f '(x) is increasing - f(x) is...
y' = -1/(1 + x²)
y' = cos(x)
? v(1 + (dy/dx)²) dx over interval a to b
concave up

2. Limit comparison test
? f(x) dx on interval a to b = F(b) - F(a)
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
1/2 ? r² over interval from a to b - find a & b by setting r = 0 - solve for theta
if lim as n approaches 8 of ratio of comparison series/general term is positive and finite - then series behaves like comparison series

3. Given velocity vectors dx/dt and dy/dt - find speed
Slope of secant line between two points - use to estimate instantanous rate of change at a point.
v(dx/dt)² + (dy/dt)² not an integral!
a^x ln(a)
positive

4. Given velocity vectors dx/dt and dy/dt - find total distance travelled
? v (dx/dt)² + (dy/dt)² over interval from a to b
draw short segments representing slope at each point
y' = -sin(x)
substitution - parts - partial fractions

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

6. p-series test
? v(t) over interval a to b
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
general term = 1/n^p - converges if p > 1

7. Formal definition of derivative
1/(b-a) ? f(x) dx on interval a to b
Limit as h approaches 0 of [f(a+h)-f(a)]/h
? f(x) dx on interval a to b = F(b) - F(a)
has limits a & b - find antiderivative - F(b) - F(a)

8. Product Rule

9. Area between two curves
use ratio test - set > 1 and solve absolute value equations - check endpoints
uv' + vu'
? f(x) - g(x) over interval a to b - where f(x) is top function and g(x) is bottom function
a^x ln(a)

10. area below x-axis is...
negative
point of inflection
concave up
use ratio test - set > 1 and solve absolute value equations - radius = center - endpoint

11. average value of f(x)
f(x) has a relative maximum
y' = 1/(1 + x²)
1/(b-a) ? f(x) dx on interval a to b
use tangent line to approximate values of the function

12. y = ln(x)/x² - state rule used to find derivative
quotient rule
1/(b-a) ? f(x) dx on interval a to b
negative
positive

13. Instantenous Rate of Change
Slope of tangent line at a point - value of derivative at a point
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
? A(x) dx over interval a to b - where A(x) is the area of the given cross-section in terms of x

14. When is a function not differentiable
point of inflection
Slope of secant line between two points - use to estimate instantanous rate of change at a point.
corner - cusp - vertical tangent - discontinuity
decreasing

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

16. [(h1 - h2)/2]*base
Area of trapezoid
? v(1 + (dy/dx)²) dx over interval a to b
negative
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)

17. Area inside polar curve
1/2 ? r² over interval from a to b - find a & b by setting r = 0 - solve for theta
corner - cusp - vertical tangent - discontinuity
? v(t) over interval a to b
general term = a1r^n - converges if -1 < r < 1

18. Volume of solid of revolution - washer
separate variables - integrate + C - use initial condition to find C - solve for y
f '(g(x)) g'(x)
y' = sec²(x)
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

19. definite integral
f(x) has a relative minimum
has limits a & b - find antiderivative - F(b) - F(a)
integrand is a rational function with a factorable denominator
concave down

20. Find radius of convergence
use ratio test - set > 1 and solve absolute value equations - radius = center - endpoint
logistic differential equation - M = carrying capacity
concave up
y' = sec²(x)

21. trapezoidal rule
velocity is negative
f(x) has a relative minimum
separate variables - integrate + C - use initial condition to find C - solve for y
use trapezoids to evaluate integrals (estimate area)

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

23. rate
derivative
Area of trapezoid
f '(g(x)) g'(x)
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)

24. To find absolute maximum on closed interval [a - b] - you must consider...
substitution - parts - partial fractions
f(x) has a relative maximum
critical points and endpoints
Limit as h approaches 0 of [f(a+h)-f(a)]/h

25. Length of curve
chain rule
derivative
? v(1 + (dy/dx)²) dx over interval a to b
polynomial with finite number of terms - largest exponent is 6 - find all derivatives up to the 6th derivative

26. mean value theorem

27. right riemann sum
use to find indeterminate limits - find derivative of numerator and denominator separately then evaluate limit
relative maximum
use rectangles with right-endpoints to evaluate integrals (estimate area)
e^x

28. Length of parametric curve
? v (dx/dt)² + (dy/dt)² over interval from a to b
use tangent line to approximate values of the function
uv' + vu'
polynomial with finite number of terms - largest exponent is 6 - find all derivatives up to the 6th derivative

29. If g(x) = ? f(t) dt on interval 2 to x - then g'(x) =...
? abs[v(t)] over interval a to b
f(x)
y' = 1/x
(uv'-vu')/v²

30. Converges conditionally
Alternating series converges and general term converges with another test
Alternating series converges and general term diverges with another test
? f(x) dx on interval a to b = F(b) - F(a)
1/(b-a) ? f(x) dx on interval a to b

31. Converges absolutely
logistic growth equation
product rule
uv' + vu'
Alternating series converges and general term converges with another test

32. y = cos²(3x)
? abs[v(t)] over interval a to b
chain rule
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.
critical points and endpoints

33. methods of integration
f(x)
derivative
zero
substitution - parts - partial fractions

34. Fundamental Theorem of Calculus
relative minimum
use to find indeterminate limits - find derivative of numerator and denominator separately then evaluate limit
? f(x) dx on interval a to b = F(b) - F(a)
undefined

35. slope of horizontal line
quotient rule
a^x ln(a)
lim as n approaches zero of general term = 0 and terms decrease - series converges
zero

36. P = M / (1 + Ae^(-Mkt))
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
logistic growth equation
draw short segments representing slope at each point
? f(x) - g(x) over interval a to b - where f(x) is top function and g(x) is bottom function

37. dP/dt = kP(M - P)
? v(1 + (dy/dx)²) dx over interval a to b
y' = 1/(1 + x²)
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.
logistic differential equation - M = carrying capacity

38. To draw a slope field - plug (x -y) coordinates into differential equation...
positive
draw short segments representing slope at each point
? f(x) - g(x) over interval a to b - where f(x) is top function and g(x) is bottom function
corner - cusp - vertical tangent - discontinuity

39. Indeterminate forms
lim as n approaches 8 of ratio of (n+1) term/nth term > 1 - series converges
0/0 - 8/8 - 8*0 - 8 - 8 - 1^8 - 0° - 8°
? f(x) - g(x) over interval a to b - where f(x) is top function and g(x) is bottom function
Limit as h approaches 0 of [f(a+h)-f(a)]/h

40. y = x cos(x) - state rule used to find derivative
product rule
lim as n approaches 8 of ratio of (n+1) term/nth term > 1 - series converges
use rectangles with right-endpoints to evaluate integrals (estimate area)
chain rule

41. Volume of solid with base in the plane and given cross-section
derivative
? A(x) dx over interval a to b - where A(x) is the area of the given cross-section in terms of x
y' = sec(x)tan(x)
point of inflection

42. If f '(x) = 0 and f'(x) < 0 -...
use rectangles with left-endpoints to evaluate integral (estimate area)
concave down
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) has a relative maximum

43. indefinite integral
point of inflection
y' = 1/(x lna)
1/2 ? R² - r² over interval from a to b - find a & b by setting equations equal - solve for theta.
no limits - find antiderivative + C - use inital value to find C

44. Find interval of convergence
use ratio test - set > 1 and solve absolute value equations - check endpoints
find first derivative - dy/dx = dy/dt / dx/dt - then find derivative of first derivative - then divide by dx/dt
relative minimum
use to find indeterminate limits - find derivative of numerator and denominator separately then evaluate limit

45. When f '(x) is positive - f(x) is...
f(x) has a relative minimum
increasing
velocity is negative
logistic growth equation

46. y = tan?¹(x) - y' =

47. When f '(x) changes from negative to positive - f(x) has a...
concave down
if lim as n approaches 8 of ratio of comparison series/general term is positive and finite - then series behaves like comparison series
relative minimum
Slope of secant line between two points - use to estimate instantanous rate of change at a point.

48. Particle is moving to the left/down
velocity is negative
? v (dx/dt)² + (dy/dt)² over interval from a to b
1/2 ? R² - r² over interval from a to b - find a & b by setting equations equal - solve for theta.
y' = sec²(x)

49. Alternating series tes
lim as n approaches zero of general term = 0 and terms decrease - series converges
critical points and endpoints
Limit as x approaches a of [f(x)-f(a)]/(x-a)
find first derivative - dy/dx = dy/dt / dx/dt - then find derivative of first derivative - then divide by dx/dt

50. Linearization
use tangent line to approximate values of the function
? v (dx/dt)² + (dy/dt)² over interval from a to b
? A(x) dx over interval a to b - where A(x) is the area of the given cross-section in terms of x
Alternating series converges and general term diverges with another test