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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 = ln(x) - y' =

2. Limit comparison test
if lim as n approaches 8 of ratio of comparison series/general term is positive and finite - then series behaves like comparison series
quotient rule
f(x) has a relative maximum
f(x)

3. 6th degree Taylor Polynomial
use ratio test - set > 1 and solve absolute value equations - check endpoints
use rectangles with left-endpoints to evaluate integral (estimate area)
y' = sec(x)tan(x)
polynomial with finite number of terms - largest exponent is 6 - find all derivatives up to the 6th derivative

4. Quotient Rule

5. area under a curve
negative
y' = 1/x
? f(x) dx integrate over interval a to b
uv' + vu'

6. average value of 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
A function and it's derivative are in the integrand
Slope of tangent line at a point - value of derivative at a point
1/(b-a) ? f(x) dx on interval a to b

7. Integral test
substitution - parts - partial fractions
if integral converges - series converges
Slope of tangent line at a point - value of derivative at a point
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)

8. Instantenous Rate of Change
y' = -1/(1 + x²)
Slope of tangent line at a point - value of derivative at a point
relative maximum
product rule

9. Converges conditionally
increasing
y' = -sin(x)
quotient rule
Alternating series converges and general term diverges with another test

10. ? u dv =
use tangent line to approximate values of the function
uv - ? v du
y' = cos(x)
? f(x) dx on interval a to b = F(b) - F(a)

11. Particle is moving to the right/up
? f(x) dx on interval a to b = F(b) - F(a)
integrand is a rational function with a factorable denominator
velocity is positive
y' = sec²(x)

12. Product Rule

13. If f '(x) = 0 and f'(x) < 0 -...
f(x) has a relative maximum
use trapezoids to evaluate integrals (estimate area)
0/0 - 8/8 - 8*0 - 8 - 8 - 1^8 - 0° - 8°
uv' + vu'

14. y = cot?¹(x) - y' =

15. When f '(x) is negative - f(x) is...
y' = 1/(1 + x²)
decreasing
concave down
Alternating series converges and general term diverges with another test

16. Area between two curves
positive
? f(x) - g(x) over interval a to b - where f(x) is top function and g(x) is bottom function
e^x
point of inflection

17. Given v(t) find displacement
0/0 - 8/8 - 8*0 - 8 - 8 - 1^8 - 0° - 8°
? v(t) over interval a to b
relative maximum
? f(x) - g(x) over interval a to b - where f(x) is top function and g(x) is bottom function

18. Average Rate of Change
? f(x) - g(x) over interval a to b - where f(x) is top function and g(x) is bottom function
speed
integrand is a rational function with a factorable denominator
Slope of secant line between two points - use to estimate instantanous rate of change at a point.

19. slope of horizontal line
? abs[v(t)] over interval a to b
use rectangles with left-endpoints to evaluate integral (estimate area)
zero
Slope of tangent line at a point - value of derivative at a point

20. When f '(x) is decreasing - f(x) is...
concave down
p ? r² dx over interval a to b - where r = distance from curve to axis of revolution
use rectangles with left-endpoints to evaluate integral (estimate area)
1/(b-a) ? f(x) dx on interval a to b

21. absolute value of velocity
y' = -csc(x)cot(x)
speed
Slope of secant line between two points - use to estimate instantanous rate of change at a 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

22. mean value theorem

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

24. Taylor series
uv - ? v du
polynomial with infinite number of terms - includes general term
decreasing
velocity is positive

25. 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.
1/2 ? R² - r² over interval from a to b - find a & b by setting equations equal - solve for theta.
lim as n approaches zero of general term = 0 and terms decrease - series converges
y' = cos(x)

26. [(h1 - h2)/2]*base
Area of trapezoid
use rectangles with right-endpoints to evaluate integrals (estimate area)
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)
? f(x) dx on interval a to b = F(b) - F(a)

27. dP/dt = kP(M - P)
f(x) has a relative maximum
use tangent line to approximate values of the function
f(x) has a relative minimum
logistic differential equation - M = carrying capacity

28. Fundamental Theorem of Calculus
? f(x) dx on interval a to b = F(b) - F(a)
polynomial with finite number of terms - largest exponent is 6 - find all derivatives up to the 6th derivative
(uv'-vu')/v²
polynomial with infinite number of terms - includes general term

29. slope of vertical line
has limits a & b - find antiderivative - F(b) - F(a)
? v (dx/dt)² + (dy/dt)² over interval from a to b
y' = -sin(x)
undefined

30. If g(x) = ? f(t) dt on interval 2 to x - then g'(x) =...
concave up
f(x)
quotient rule
zero

31. Volume of solid of revolution - no washer
draw short segments representing slope at each point
p ? r² dx over interval a to b - where r = distance from curve to axis of revolution
Slope of secant line between two points - use to estimate instantanous rate of change at a point.
Limit as x approaches a of [f(x)-f(a)]/(x-a)

32. If f '(x) = 0 and f'(x) > 0 -...
(uv'-vu')/v²
concave down
f(x) has a relative minimum
? abs[v(t)] over interval a to b

33. y = tan?¹(x) - y' =

34. Given velocity vectors dx/dt and dy/dt - find total distance travelled
zero
negative
? v (dx/dt)² + (dy/dt)² over interval from a to b
use to find indeterminate limits - find derivative of numerator and denominator separately then evaluate limit

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

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

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

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

39. L'Hopitals rule
? v (dx/dt)² + (dy/dt)² over interval from a to b
use to find indeterminate limits - find derivative of numerator and denominator separately then evaluate limit
find first derivative - dy/dx = dy/dt / dx/dt - then find derivative of first derivative - then divide by dx/dt
Slope of secant line between two points - use to estimate instantanous rate of change at a point.

40. right riemann sum
y' = -csc²(x)
use ratio test - set > 1 and solve absolute value equations - check endpoints
1/2 ? R² - r² over interval from a to b - find a & b by setting equations equal - solve for theta.
use rectangles with right-endpoints to evaluate integrals (estimate area)

41. When f '(x) changes from negative to positive - f(x) has a...
e^x
relative minimum
chain rule
Slope of tangent line at a point - value of derivative at a point

42. rate
Slope of tangent line at a point - value of derivative at a point
use to find indeterminate limits - find derivative of numerator and denominator separately then evaluate limit
derivative
velocity is negative

43. p-series test
? v (dx/dt)² + (dy/dt)² over interval from a to b
y' = sec(x)tan(x)
1/(b-a) ? f(x) dx on interval a to b
general term = 1/n^p - converges if p > 1

44. To find particular solution to differential equation - dy/dx = x/y...
separate variables - integrate + C - use initial condition to find C - solve for y
1/(b-a) ? f(x) dx on interval a to b
logistic differential equation - M = carrying capacity
Limit as x approaches a of [f(x)-f(a)]/(x-a)

45. Length of curve
has limits a & b - find antiderivative - F(b) - F(a)
uv' + vu'
? f(x) dx integrate over interval a to b
? v(1 + (dy/dx)²) dx over interval a to b

46. Length of parametric curve
lim as n approaches zero of general term = 0 and terms decrease - series converges
positive
? 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

47. To draw a slope field - plug (x -y) coordinates into differential equation...
general term = a1r^n - converges if -1 < r < 1
use ratio test - set > 1 and solve absolute value equations - radius = center - endpoint
draw short segments representing slope at each point
f(x) has a relative maximum

48. trapezoidal rule
logistic differential equation - M = carrying capacity
separate variables - integrate + C - use initial condition to find C - solve for y
use trapezoids to evaluate integrals (estimate area)
use tangent line to approximate values of the function

49. y = ln(x)/x² - state rule used to find derivative
f(x)
y' = -1/(1 + x²)
relative minimum
quotient rule

50. Alternate definition of derivative
velocity is negative
separate variables - integrate + C - use initial condition to find C - solve for y
if terms grow without bound - series diverges
Limit as x approaches a of [f(x)-f(a)]/(x-a)