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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. Formal definition of derivative
y' = cos(x)
two different types of functions are multiplied
Limit as h approaches 0 of [f(a+h)-f(a)]/h
use tangent line to approximate values of the function

2. use integration by parts when...
draw short segments representing slope at each point
two different types of functions are multiplied
product rule
y' = 1/v(1 - x²)

3. If f '(x) = 0 and f'(x) > 0 -...
polynomial with finite number of terms - largest exponent is 6 - find all derivatives up to the 6th derivative
increasing
? f(x) dx on interval a to b = F(b) - F(a)
f(x) has a relative minimum

4. y = ln(x)/x² - state rule used to find derivative
quotient rule
relative maximum
? v(1 + (dy/dx)²) dx over interval a to b
point of inflection

5. ? u dv =
y' = -1/(1 + x²)
v(dx/dt)² + (dy/dt)² not an integral!
uv - ? v du
p ? r² dx over interval a to b - where r = distance from curve to axis of revolution

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

7. y = cos²(3x)
chain rule
no limits - find antiderivative + C - use inital value to find C
velocity is positive
corner - cusp - vertical tangent - discontinuity

8. Converges conditionally
1/2 ? R² - r² over interval from a to b - find a & b by setting equations equal - solve for theta.
negative
Alternating series converges and general term diverges with another test
use rectangles with left-endpoints to evaluate integral (estimate area)

9. To find particular solution to differential equation - dy/dx = x/y...
? v(t) over interval a to b
separate variables - integrate + C - use initial condition to find C - solve for y
? f(x) - g(x) over interval a to b - where f(x) is top function and g(x) is bottom function
y' = 1/x

10. Integral test
Alternating series converges and general term converges with another test
if integral converges - series converges
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.
(uv'-vu')/v²

11. Length of curve
? v (dx/dt)² + (dy/dt)² over interval from a to b
? v(1 + (dy/dx)²) dx over interval a to b
? v(t) over interval a to b
f(x) has a relative maximum

12. Use partial fractions to integrate when...
p ? r² dx over interval a to b - where r = distance from curve to axis of revolution
use ratio test - set > 1 and solve absolute value equations - radius = center - endpoint
integrand is a rational function with a factorable denominator
1/2 ? R² - r² over interval from a to b - find a & b by setting equations equal - solve for theta.

13. Taylor series
substitution - parts - partial fractions
velocity is positive
polynomial with infinite number of terms - includes general term
separate variables - integrate + C - use initial condition to find C - solve for y

14. methods of integration
speed
general term = a1r^n - converges if -1 < r < 1
? f(x) - g(x) over interval a to b - where f(x) is top function and g(x) is bottom function
substitution - parts - partial fractions

15. nth term test
separate variables - integrate + C - use initial condition to find C - solve for y
relative maximum
no limits - find antiderivative + C - use inital value to find C
if terms grow without bound - series diverges

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

17. y = cot?¹(x) - y' =

18. Volume of solid of revolution - washer
Alternating series converges and general term diverges with another test
v(dx/dt)² + (dy/dt)² not an integral!
? v(t) over interval 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

19. Average Rate of Change
concave up
Slope of secant line between two points - use to estimate instantanous rate of change at a point.
undefined
general term = a1r^n - converges if -1 < r < 1

20. Particle is moving to the right/up
velocity is positive
f(x) has a relative maximum
logistic growth equation
e^x

21. y = a^x - y' = y' =
relative maximum
a^x ln(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
v(dx/dt)² + (dy/dt)² not an integral!

22. Given v(t) find displacement
logistic growth equation
? v(t) over interval a to b
y' = -1/(1 + x²)
A function and it's derivative are in the integrand

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

24. Chain Rule

25. Geometric series test
concave up
Slope of secant line between two points - use to estimate instantanous rate of change at a point.
general term = a1r^n - converges if -1 < r < 1
Limit as x approaches a of [f(x)-f(a)]/(x-a)

26. Alternating series tes
? v (dx/dt)² + (dy/dt)² over interval from a to b
if integral converges - series converges
use rectangles with left-endpoints to evaluate integral (estimate area)
lim as n approaches zero of general term = 0 and terms decrease - series converges

27. When f '(x) is increasing - f(x) is...
concave up
use rectangles with left-endpoints to evaluate integral (estimate area)
? A(x) dx over interval a to b - where A(x) is the area of the given cross-section in terms of x
uv - ? v du

28. Limit comparison test
velocity is negative
if lim as n approaches 8 of ratio of comparison series/general term is positive and finite - then series behaves like comparison series
if terms grow without bound - series diverges
lim as n approaches zero of general term = 0 and terms decrease - series converges

29. slope of vertical line
undefined
use ratio test - set > 1 and solve absolute value equations - check endpoints
? v (dx/dt)² + (dy/dt)² over interval from a to b
point of inflection

30. trapezoidal rule
critical points and endpoints
use trapezoids to evaluate integrals (estimate area)
y' = sec(x)tan(x)
y' = 1/(1 + x²)

31. left riemann sum
chain rule
use rectangles with left-endpoints to evaluate integral (estimate area)
y' = -sin(x)
y' = sec(x)tan(x)

32. Indeterminate forms
draw short segments representing slope at each point
? v(t) over interval a to b
(uv'-vu')/v²
0/0 - 8/8 - 8*0 - 8 - 8 - 1^8 - 0° - 8°

33. If f '(x) = 0 and f'(x) < 0 -...
quotient rule
(uv'-vu')/v²
f(x) has a relative maximum
uv' + vu'

34. Alternate definition of derivative
point of inflection
y' = cos(x)
lim as n approaches zero of general term = 0 and terms decrease - series converges
Limit as x approaches a of [f(x)-f(a)]/(x-a)

35. Given v(t) find total distance travelled
has limits a & b - find antiderivative - F(b) - F(a)
? v(1 + (dy/dx)²) dx over interval a to b
? abs[v(t)] over interval a to b
use trapezoids to evaluate integrals (estimate area)

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

37. To draw a slope field - plug (x -y) coordinates into differential equation...
velocity is positive
draw short segments representing slope at each point
? v(t) over interval a to b
use tangent line to approximate values of the function

38. absolute value of velocity
speed
1/2 ? R² - r² over interval from a to b - find a & b by setting equations equal - solve for theta.
uv' + vu'
chain rule

39. y = x cos(x) - state rule used to find derivative
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)
zero
uv' + vu'
product rule

40. Given velocity vectors dx/dt and dy/dt - find total distance travelled
y' = -1/(1 + x²)
? v (dx/dt)² + (dy/dt)² over interval from a to b
substitution - parts - partial fractions
polynomial with infinite number of terms - includes general term

41. y = cos?¹(x) - y' =

42. Eatio test
lim as n approaches 8 of ratio of (n+1) term/nth term > 1 - series converges
y' = 1/v(1 - x²)
? f(x) dx integrate over interval a to b
use trapezoids to evaluate integrals (estimate area)

43. Second derivative of parametrically defined curve
f '(g(x)) g'(x)
find first derivative - dy/dx = dy/dt / dx/dt - then find derivative of first derivative - then divide by dx/dt
Slope of tangent line at a point - value of derivative at a point
0/0 - 8/8 - 8*0 - 8 - 8 - 1^8 - 0° - 8°

44. p-series test
general term = 1/n^p - converges if p > 1
derivative
negative
corner - cusp - vertical tangent - discontinuity

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

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

47. slope of horizontal line
if integral converges - series converges
y' = sec²(x)
quotient rule
zero

48. Area inside one polar curve and outside another polar curve
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' = -1/(1 + x²)
relative minimum

49. Linearization
concave up
? f(x) dx on interval a to b = F(b) - F(a)
use ratio test - set > 1 and solve absolute value equations - radius = center - endpoint
use tangent line to approximate values of the function

50. When f '(x) is decreasing - f(x) is...
concave down
? v(t) over interval a to b
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)
Alternating series converges and general term diverges with another test