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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 = tan?¹(x) - y' =

2. 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
f(x)
? A(x) dx over interval a to b - where A(x) is the area of the given cross-section in terms of x
use rectangles with right-endpoints to evaluate integrals (estimate area)

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

4. use substitution to integrate when

5. rate
derivative
separate variables - integrate + C - use initial condition to find C - solve for y
y' = 1/x
use to find indeterminate limits - find derivative of numerator and denominator separately then evaluate limit

6. area under a curve
y' = 1/x
Slope of secant line between two points - use to estimate instantanous rate of change at a point.
? f(x) dx integrate over interval a to b
zero

7. y = cos²(3x)
point of inflection
decreasing
chain rule
? f(x) dx integrate over interval a to b

8. P = M / (1 + Ae^(-Mkt))
uv - ? v du
1/(b-a) ? f(x) dx on interval a to b
polynomial with infinite number of terms - includes general term
logistic growth equation

9. When f '(x) changes from increasing to decreasing or decreasing to increasing - f(x) has a...
A function and it's derivative are in the integrand
y' = -sin(x)
point of inflection
Limit as x approaches a of [f(x)-f(a)]/(x-a)

10. L'Hopitals rule
? f(x) dx on interval a to b = F(b) - F(a)
if terms grow without bound - series diverges
substitution - parts - partial fractions
use to find indeterminate limits - find derivative of numerator and denominator separately then evaluate limit

11. Converges absolutely
Alternating series converges and general term converges with another test
lim as n approaches zero of general term = 0 and terms decrease - series converges
? abs[v(t)] over interval a to b
(uv'-vu')/v²

12. methods of integration
velocity is positive
e^x
y' = sec²(x)
substitution - parts - partial fractions

13. area below x-axis is...
polynomial with finite number of terms - largest exponent is 6 - find all derivatives up to the 6th derivative
general term = a1r^n - converges if -1 < r < 1
negative
chain rule

14. Given v(t) find total distance travelled
? abs[v(t)] over interval a to b
y' = sec²(x)
draw short segments representing slope at each point
Limit as x approaches a of [f(x)-f(a)]/(x-a)

15. Length of curve
derivative
logistic differential equation - M = carrying capacity
? v (dx/dt)² + (dy/dt)² over interval from a to b
? v(1 + (dy/dx)²) dx over interval a to b

16. 6th degree Taylor Polynomial
polynomial with finite number of terms - largest exponent is 6 - find all derivatives up to the 6th derivative
polynomial with infinite number of terms - includes general term
two different types of functions are multiplied
Alternating series converges and general term converges with another test

17. Average Rate of Change
Alternating series converges and general term diverges with another test
Slope of secant line between two points - use to estimate instantanous rate of change at a point.
y' = -csc²(x)
0/0 - 8/8 - 8*0 - 8 - 8 - 1^8 - 0° - 8°

18. trapezoidal rule
y' = -1/(1 + x²)
use trapezoids to evaluate integrals (estimate area)
relative minimum
Limit as h approaches 0 of [f(a+h)-f(a)]/h

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

20. To find absolute maximum on closed interval [a - b] - you must consider...
y' = 1/(x lna)
y' = 1/v(1 - x²)
lim as n approaches zero of general term = 0 and terms decrease - series converges
critical points and endpoints

21. If f '(x) = 0 and f'(x) > 0 -...
A function and it's derivative are in the integrand
f(x) has a relative minimum
velocity is positive
y' = 1/(x lna)

22. Given velocity vectors dx/dt and dy/dt - find speed
v(dx/dt)² + (dy/dt)² not an integral!
corner - cusp - vertical tangent - discontinuity
(uv'-vu')/v²
logistic growth equation

23. y = cot?¹(x) - y' =

24. y = x cos(x) - state rule used to find derivative
product rule
Limit as x approaches a of [f(x)-f(a)]/(x-a)
positive
y' = cos(x)

25. left riemann sum
use rectangles with left-endpoints to evaluate integral (estimate area)
use tangent line to approximate values of the function
if integral converges - series converges
concave down

26. [(h1 - h2)/2]*base
? v(t) over interval a to b
Area of trapezoid
product rule
y' = 1/(x lna)

27. Chain Rule

28. Volume of solid with base in the plane and given cross-section
y' = cos(x)
relative maximum
Alternating series converges and general term converges with another test
? A(x) dx over interval a to b - where A(x) is the area of the given cross-section in terms of x

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

30. When is a function not differentiable
uv' + vu'
corner - cusp - vertical tangent - discontinuity
y' = -1/v(1 - x²)
uv - ? v du

31. Area between two curves
uv - ? v du
use rectangles with left-endpoints to evaluate integral (estimate area)
substitution - parts - partial fractions
? f(x) - g(x) over interval a to b - where f(x) is top function and g(x) is bottom function

32. Particle is moving to the right/up
f '(g(x)) g'(x)
a^x ln(a)
velocity is positive
separate variables - integrate + C - use initial condition to find C - solve for y

33. When f '(x) changes from negative to positive - f(x) has 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
y' = -1/(1 + x²)
relative minimum
a^x ln(a)

34. absolute value of velocity
critical points and endpoints
derivative
general term = a1r^n - converges if -1 < r < 1
speed

35. When f '(x) is negative - f(x) is...
? f(x) - g(x) over interval a to b - where f(x) is top function and g(x) is bottom function
(uv'-vu')/v²
speed
decreasing

36. average value of f(x)
y' = cos(x)
find first derivative - dy/dx = dy/dt / dx/dt - then find derivative of first derivative - then divide by dx/dt
uv - ? v du
1/(b-a) ? f(x) dx on interval a to b

37. Find radius of convergence
y' = -sin(x)
use ratio test - set > 1 and solve absolute value equations - radius = center - endpoint
1/(b-a) ? f(x) dx on interval a to b
polynomial with infinite number of terms - includes general term

38. dP/dt = kP(M - P)
logistic differential equation - M = carrying capacity
use trapezoids to evaluate integrals (estimate area)
1/2 ? r² over interval from a to b - find a & b by setting r = 0 - solve for theta
uv' + vu'

39. Taylor series
product rule
lim as n approaches zero of general term = 0 and terms decrease - series converges
polynomial with infinite number of terms - includes general term
(uv'-vu')/v²

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

41. If f '(x) = 0 and f'(x) < 0 -...
use ratio test - set > 1 and solve absolute value equations - check endpoints
f(x) has a relative maximum
use rectangles with left-endpoints to evaluate integral (estimate area)
use tangent line to approximate values of the function

42. Given velocity vectors dx/dt and dy/dt - find total distance travelled
A function and it's derivative are in the integrand
? 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
has limits a & b - find antiderivative - F(b) - F(a)

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

44. y = e^x - y' = y' =
e^x
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.
chain rule

45. y = cos?¹(x) - y' =

46. Product Rule

47. use integration by parts when...
two different types of functions are multiplied
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.
y' = sec²(x)

48. y = ln(x)/x² - state rule used to find derivative
y' = -1/(1 + x²)
(uv'-vu')/v²
velocity is positive
quotient rule

49. Volume of solid of revolution - no washer
p ? r² dx over interval a to b - where r = distance from curve to axis of revolution
derivative
1/(b-a) ? f(x) dx on interval a to b
? f(x) dx on interval a to b = F(b) - F(a)

50. mean value theorem