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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. Particle is moving to the right/up
velocity is positive
logistic differential equation - M = carrying capacity
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

2. When f '(x) changes from increasing to decreasing or decreasing to increasing - f(x) has a...
y' = 1/v(1 - x²)
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)
uv' + vu'
point of inflection

3. Length of curve
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
? v(1 + (dy/dx)²) dx over interval a to b
increasing

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

5. ? u dv =
concave up
quotient rule
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)
uv - ? v du

6. right riemann sum
p ? r² dx over interval a to b - where r = distance from 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
use rectangles with right-endpoints to evaluate integrals (estimate area)
y' = -1/(1 + x²)

7. left riemann sum
? v(1 + (dy/dx)²) dx over interval a to b
lim as n approaches zero of general term = 0 and terms decrease - series converges
use rectangles with left-endpoints to evaluate integral (estimate area)
f(x) has a relative maximum

8. y = tan?¹(x) - y' =

9. rate
y' = -csc(x)cot(x)
derivative
y' = -1/(1 + x²)
negative

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

11. When f '(x) is negative - f(x) is...
general term = a1r^n - converges if -1 < r < 1
two different types of functions are multiplied
use rectangles with right-endpoints to evaluate integrals (estimate area)
decreasing

12. Given velocity vectors dx/dt and dy/dt - find total distance travelled
if lim as n approaches 8 of ratio of comparison series/general term is positive and finite - then series behaves like comparison series
Slope of tangent line at a point - value of derivative at a point
? v(t) over interval a to b
? v (dx/dt)² + (dy/dt)² over interval from a to b

13. Volume of solid of revolution - no washer
? A(x) dx over interval a to b - where A(x) is the area of the given cross-section in terms of x
p ? r² dx over interval a to b - where r = distance from curve to axis of revolution
y' = -1/(1 + x²)
chain rule

14. trapezoidal rule
no limits - find antiderivative + C - use inital value to find C
velocity is negative
draw short segments representing slope at each point
use trapezoids to evaluate integrals (estimate area)

15. Second derivative of parametrically defined curve
speed
find first derivative - dy/dx = dy/dt / dx/dt - then find derivative of first derivative - then divide by dx/dt
undefined
decreasing

16. Geometric series test
0/0 - 8/8 - 8*0 - 8 - 8 - 1^8 - 0° - 8°
e^x
general term = a1r^n - converges if -1 < r < 1
A function and it's derivative are in the integrand

17. Length of parametric curve
y' = -csc(x)cot(x)
has limits a & b - find antiderivative - F(b) - F(a)
? v (dx/dt)² + (dy/dt)² over interval from a to b
positive

18. y = e^x - y' = y' =
y' = 1/x
e^x
y' = -sin(x)
Slope of tangent line at a point - value of derivative at a point

19. When f '(x) changes from negative to positive - f(x) has a...
Area of trapezoid
relative minimum
y' = -sin(x)
y' = 1/(x lna)

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

21. use substitution to integrate when

22. methods of integration
polynomial with infinite number of terms - includes general term
substitution - parts - partial fractions
Alternating series converges and general term diverges with another test
zero

23. Area inside one polar curve and outside another polar curve
polynomial with finite number of terms - largest exponent is 6 - find all derivatives up to the 6th derivative
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.
Slope of secant line between two points - use to estimate instantanous rate of change at a point.

24. y = cos²(3x)
polynomial with infinite number of terms - includes general term
y' = 1/v(1 - x²)
v(dx/dt)² + (dy/dt)² not an integral!
chain rule

25. y = x cos(x) - state rule used to find derivative
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
product rule
(uv'-vu')/v²

26. slope of vertical line
y' = -csc(x)cot(x)
y' = sec(x)tan(x)
undefined
substitution - parts - partial fractions

27. average value of f(x)
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
(uv'-vu')/v²
y' = -1/v(1 - x²)

28. P = M / (1 + Ae^(-Mkt))
logistic growth equation
chain rule
Limit as h approaches 0 of [f(a+h)-f(a)]/h
use rectangles with left-endpoints to evaluate integral (estimate area)

29. Area between two curves
Limit as x approaches a of [f(x)-f(a)]/(x-a)
? f(x) - g(x) over interval a to b - where f(x) is top function and g(x) is bottom function
relative maximum
? abs[v(t)] over interval a to b

30. If g(x) = ? f(t) dt on interval 2 to x - then g'(x) =...
f(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
use ratio test - set > 1 and solve absolute value equations - check endpoints
relative maximum

31. slope of horizontal line
zero
use rectangles with right-endpoints to evaluate integrals (estimate area)
use ratio test - set > 1 and solve absolute value equations - check endpoints
uv' + vu'

32. use integration by parts when...
y' = -sin(x)
use trapezoids to evaluate integrals (estimate area)
two different types of functions are multiplied
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

33. y = a^x - y' = y' =
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.
a^x ln(a)
velocity is negative
f '(g(x)) g'(x)

34. Given velocity vectors dx/dt and dy/dt - find speed
polynomial with finite number of terms - largest exponent is 6 - find all derivatives up to the 6th derivative
v(dx/dt)² + (dy/dt)² not an integral!
? abs[v(t)] over interval a to b
y' = -csc(x)cot(x)

35. Chain Rule

36. Volume of solid with base in the plane and given cross-section
undefined
? f(x) dx on interval a to b = F(b) - F(a)
velocity is negative
? A(x) dx over interval a to b - where A(x) is the area of the given cross-section in terms of x

37. Limit comparison test
general term = 1/n^p - converges if p > 1
? A(x) dx over interval a to b - where A(x) is the area of the given cross-section in terms of x
if lim as n approaches 8 of ratio of comparison series/general term is positive and finite - then series behaves like comparison series
point of inflection

38. Product Rule

39. When f '(x) changes fro positive to negative - f(x) has a...
find first derivative - dy/dx = dy/dt / dx/dt - then find derivative of first derivative - then divide by dx/dt
relative maximum
? f(x) - g(x) over interval a to b - where f(x) is top function and g(x) is bottom function
point of inflection

40. Alternate definition of derivative
Limit as x approaches a of [f(x)-f(a)]/(x-a)
uv' + vu'
y' = 1/x
Alternating series converges and general term converges with another test

41. Use partial fractions to integrate when...
velocity is negative
(uv'-vu')/v²
Limit as x approaches a of [f(x)-f(a)]/(x-a)
integrand is a rational function with a factorable denominator

42. To draw a slope field - plug (x -y) coordinates into differential equation...
draw short segments representing slope at each point
? v (dx/dt)² + (dy/dt)² over interval from a to b
uv' + vu'
? f(x) dx integrate over interval a to b

43. Converges conditionally
Alternating series converges and general term diverges with another test
1/2 ? r² over interval from a to b - find a & b by setting r = 0 - solve for theta
no limits - find antiderivative + C - use inital value to find C
use ratio test - set > 1 and solve absolute value equations - check endpoints

44. [(h1 - h2)/2]*base
quotient rule
e^x
polynomial with finite number of terms - largest exponent is 6 - find all derivatives up to the 6th derivative
Area of trapezoid

45. Intermediate Value Theorem
derivative
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.
1/2 ? r² over interval from a to b - find a & b by setting r = 0 - solve for theta
zero

46. Converges absolutely
Alternating series converges and general term diverges with another test
? abs[v(t)] over interval a to b
Alternating series converges and general term converges with another test
Area of trapezoid

47. When f '(x) is positive - f(x) is...
integrand is a rational function with a factorable denominator
zero
increasing
Slope of tangent line at a point - value of derivative at a point

48. y = cot?¹(x) - y' =

49. Volume of solid of revolution - washer
polynomial with finite number of terms - largest exponent is 6 - find all derivatives up to the 6th derivative
? v (dx/dt)² + (dy/dt)² over interval from a to b
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
e^x

50. When is a function not differentiable
corner - cusp - vertical tangent - discontinuity
use rectangles with right-endpoints to evaluate integrals (estimate area)
y' = 1/v(1 - x²)
f(x) has a relative maximum