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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 = sin(x) - y' =

2. Limit comparison test
critical points and endpoints
lim as n approaches zero of general term = 0 and terms decrease - series converges
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)
if lim as n approaches 8 of ratio of comparison series/general term is positive and finite - then series behaves like comparison series

3. Volume of solid of revolution - washer
use ratio test - set > 1 and solve absolute value equations - radius = center - endpoint
positive
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
separate variables - integrate + C - use initial condition to find C - solve for y

4. Particle is moving to the right/up
velocity is positive
product rule
y' = 1/x
Limit as x approaches a of [f(x)-f(a)]/(x-a)

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

6. area under a curve
positive
uv' + vu'
? f(x) dx integrate over interval a to b
polynomial with infinite number of terms - includes general term

7. methods of integration
concave up
substitution - parts - partial fractions
1/2 ? r² over interval from a to b - find a & b by setting r = 0 - solve for theta
use ratio test - set > 1 and solve absolute value equations - radius = center - endpoint

8. Second derivative of parametrically defined curve
find first derivative - dy/dx = dy/dt / dx/dt - then find derivative of first derivative - then divide by dx/dt
use to find indeterminate limits - find derivative of numerator and denominator separately then evaluate limit
A function and it's derivative are in the integrand
lim as n approaches zero of general term = 0 and terms decrease - series converges

9. L'Hopitals rule
use to find indeterminate limits - find derivative of numerator and denominator separately then evaluate limit
increasing
Area of trapezoid
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)

10. Alternating series tes
y' = 1/v(1 - x²)
y' = -1/(1 + x²)
lim as n approaches zero of general term = 0 and terms decrease - series converges
quotient rule

11. area below x-axis is...
negative
y' = -csc(x)cot(x)
e^x
Slope of tangent line at a point - value of derivative at a point

12. When f '(x) changes from negative to positive - f(x) has a...
(uv'-vu')/v²
relative maximum
relative minimum
logistic differential equation - M = carrying capacity

13. Use partial fractions to integrate when...
y' = -sin(x)
corner - cusp - vertical tangent - discontinuity
integrand is a rational function with a factorable denominator
use to find indeterminate limits - find derivative of numerator and denominator separately then evaluate limit

14. Linearization
general term = a1r^n - converges if -1 < r < 1
? 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

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

16. y = e^x - y' = y' =
e^x
0/0 - 8/8 - 8*0 - 8 - 8 - 1^8 - 0° - 8°
two different types of functions are multiplied
integrand is a rational function with a factorable denominator

17. Formal definition of derivative
Limit as h approaches 0 of [f(a+h)-f(a)]/h
use tangent line to approximate values of the function
find first derivative - dy/dx = dy/dt / dx/dt - then find derivative of first derivative - then divide by dx/dt
y' = 1/x

18. When f '(x) changes fro positive to negative - f(x) has a...
relative maximum
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
1/(b-a) ? f(x) dx on interval a to b

19. y = sin?¹(x) - y' =

20. average value of f(x)
y' = 1/x
1/(b-a) ? f(x) dx on interval a to b
has limits a & b - find antiderivative - F(b) - F(a)
use to find indeterminate limits - find derivative of numerator and denominator separately then evaluate limit

21. Product Rule

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

23. Volume of solid with base in the plane and given cross-section
? A(x) dx over interval a to b - where A(x) is the area of the given cross-section in terms of x
point of inflection
logistic differential equation - M = carrying capacity
Limit as h approaches 0 of [f(a+h)-f(a)]/h

24. Indeterminate forms
chain rule
? v(1 + (dy/dx)²) dx over interval a to b
0/0 - 8/8 - 8*0 - 8 - 8 - 1^8 - 0° - 8°
integrand is a rational function with a factorable denominator

25. Chain Rule

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

27. Integral test
if terms grow without bound - series diverges
velocity is negative
Alternating series converges and general term diverges with another test
if integral converges - series converges

28. dP/dt = kP(M - P)
y' = cos(x)
logistic differential equation - M = carrying capacity
1/2 ? R² - r² over interval from a to b - find a & b by setting equations equal - solve for theta.
e^x

29. Given velocity vectors dx/dt and dy/dt - find speed
v(dx/dt)² + (dy/dt)² not an integral!
velocity is negative
a^x ln(a)
use ratio test - set > 1 and solve absolute value equations - check endpoints

30. Geometric series test
use rectangles with right-endpoints to evaluate integrals (estimate area)
draw short segments representing slope at each point
v(dx/dt)² + (dy/dt)² not an integral!
general term = a1r^n - converges if -1 < r < 1

31. y = cos?¹(x) - y' =

32. trapezoidal rule
use trapezoids to evaluate integrals (estimate area)
? v(1 + (dy/dx)²) dx over interval a to b
Limit as x approaches a of [f(x)-f(a)]/(x-a)
f '(g(x)) g'(x)

33. Average Rate of Change
concave up
f '(g(x)) g'(x)
critical points and endpoints
Slope of secant line between two points - use to estimate instantanous rate of change at a point.

34. Length of parametric curve
p ? r² dx over interval a to b - where r = distance from curve to axis of revolution
quotient rule
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

35. If f '(x) = 0 and f'(x) < 0 -...
A function and it's derivative are in the integrand
f(x) has a relative minimum
f(x) has a relative maximum
p ? r² dx over interval a to b - where r = distance from curve to axis of revolution

36. When f '(x) is decreasing - f(x) is...
use ratio test - set > 1 and solve absolute value equations - check endpoints
? abs[v(t)] over interval a to b
concave down
f(x) has a relative minimum

37. y = cot?¹(x) - y' =

38. When is a function not differentiable
corner - cusp - vertical tangent - discontinuity
chain rule
y' = sec²(x)
integrand is a rational function with a factorable denominator

39. nth term test
y' = -1/v(1 - x²)
? abs[v(t)] over interval a to b
separate variables - integrate + C - use initial condition to find C - solve for y
if terms grow without bound - series diverges

40. Particle is moving to the left/down
y' = 1/x
if lim as n approaches 8 of ratio of comparison series/general term is positive and finite - then series behaves like comparison series
no limits - find antiderivative + C - use inital value to find C
velocity is negative

41. p-series test
general term = 1/n^p - converges if p > 1
Slope of tangent line at a point - value of derivative at a point
if lim as n approaches 8 of ratio of comparison series/general term is positive and finite - then series behaves like comparison series
separate variables - integrate + C - use initial condition to find C - solve for y

42. Converges absolutely
y' = 1/v(1 - x²)
polynomial with finite number of terms - largest exponent is 6 - find all derivatives up to the 6th derivative
Alternating series converges and general term converges with another test
integrand is a rational function with a factorable denominator

43. Instantenous Rate of Change
Alternating series converges and general term converges with another test
f(x)
Slope of tangent line at a point - value of derivative at a point
no limits - find antiderivative + C - use inital value to find C

44. Find radius of convergence
use ratio test - set > 1 and solve absolute value equations - radius = center - endpoint
y' = -sin(x)
derivative
? v(t) over interval a to b

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

46. Length of curve
? A(x) dx over interval a to b - where A(x) is the area of the given cross-section in terms of x
? v(1 + (dy/dx)²) dx over interval a to b
y' = 1/v(1 - x²)
1/(b-a) ? f(x) dx on interval a to b

47. Eatio test
has limits a & b - find antiderivative - F(b) - F(a)
relative minimum
y' = 1/(x lna)
lim as n approaches 8 of ratio of (n+1) term/nth term > 1 - series converges

48. Area inside polar curve
? v (dx/dt)² + (dy/dt)² over interval from a to b
1/2 ? r² over interval from a to b - find a & b by setting r = 0 - solve for theta
Slope of tangent line at a point - value of derivative at a point
y' = 1/(x lna)

49. area above x-axis is...
relative minimum
zero
positive
y' = -csc²(x)

50. Given velocity vectors dx/dt and dy/dt - find total distance travelled
has limits a & b - find antiderivative - F(b) - F(a)
Alternating series converges and general term converges with another test
p ? r² dx over interval a to b - where r = distance from curve to axis of revolution
? v (dx/dt)² + (dy/dt)² over interval from a to b