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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 = a^x - y' = y' =
a^x ln(a)
chain rule
positive
use trapezoids to evaluate integrals (estimate area)

2. When f '(x) is positive - f(x) is...
? f(x) dx on interval a to b = F(b) - F(a)
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)
increasing
Area of trapezoid

3. y = x cos(x) - state rule used to find derivative
? v(1 + (dy/dx)²) dx over interval a to b
product rule
decreasing
negative

4. definite integral
uv - ? v du
has limits a & b - find antiderivative - F(b) - F(a)
if integral converges - series converges
f(x)

5. y = tan?¹(x) - y' =

6. Intermediate Value Theorem
integrand is a rational function with a factorable denominator
y' = cos(x)
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.
undefined

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

8. left riemann sum
has limits a & b - find antiderivative - F(b) - F(a)
? f(x) dx integrate over interval a to b
use rectangles with left-endpoints to evaluate integral (estimate area)
f(x) has a relative minimum

9. Volume of solid with base in the plane and given cross-section
p ? r² dx over interval a to b - where r = distance from curve to axis of revolution
? A(x) dx over interval a to b - where A(x) is the area of the given cross-section in terms of x
no limits - find antiderivative + C - use inital value to find C
1/2 ? R² - r² over interval from a to b - find a & b by setting equations equal - solve for theta.

10. When is a function not differentiable
corner - cusp - vertical tangent - discontinuity
derivative
? v(t) over interval a to b
undefined

11. rate
critical points and endpoints
derivative
y' = 1/(1 + x²)
f(x)

12. right riemann sum
Alternating series converges and general term converges with another test
use rectangles with right-endpoints to evaluate integrals (estimate area)
concave down
y' = 1/v(1 - x²)

13. Instantenous Rate of Change
draw short segments representing slope at each point
Slope of tangent line at a point - value of derivative at a point
lim as n approaches 8 of ratio of (n+1) term/nth term > 1 - series converges
negative

14. Linearization
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.
positive
use tangent line to approximate values of the function
f(x) has a relative minimum

15. use integration by parts when...
y' = 1/v(1 - x²)
two different types of functions are multiplied
y' = -1/v(1 - x²)
uv - ? v du

16. indefinite integral
no limits - find antiderivative + C - use inital value to find C
? f(x) dx integrate over interval a to b
logistic differential equation - M = carrying capacity
Limit as x approaches a of [f(x)-f(a)]/(x-a)

17. average value of f(x)
0/0 - 8/8 - 8*0 - 8 - 8 - 1^8 - 0° - 8°
? f(x) dx integrate over interval a to b
a^x ln(a)
1/(b-a) ? f(x) dx on interval a to b

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

19. absolute value of velocity
speed
lim as n approaches zero of general term = 0 and terms decrease - series converges
velocity is positive
Alternating series converges and general term converges with another test

20. Integral test
logistic differential equation - M = carrying capacity
critical points and endpoints
if integral converges - series converges
use to find indeterminate limits - find derivative of numerator and denominator separately then evaluate limit

21. Given v(t) find displacement
zero
corner - cusp - vertical tangent - discontinuity
? v(t) over interval a to b
use to find indeterminate limits - find derivative of numerator and denominator separately then evaluate limit

22. If f '(x) = 0 and f'(x) < 0 -...
f(x) has a relative maximum
has limits a & b - find antiderivative - F(b) - F(a)
if integral converges - series converges
zero

23. P = M / (1 + Ae^(-Mkt))
general term = a1r^n - converges if -1 < r < 1
logistic growth equation
if lim as n approaches 8 of ratio of comparison series/general term is positive and finite - then series behaves like comparison series
f(x) has a relative minimum

24. Geometric series test
point of inflection
general term = a1r^n - converges if -1 < r < 1
? abs[v(t)] over interval a to b
a^x ln(a)

25. Given velocity vectors dx/dt and dy/dt - find speed
1/(b-a) ? f(x) dx on interval a to b
polynomial with infinite number of terms - includes general term
v(dx/dt)² + (dy/dt)² not an integral!
draw short segments representing slope at each point

26. Area between two curves
? f(x) - g(x) over interval a to b - where f(x) is top function and g(x) is bottom function
use trapezoids to evaluate integrals (estimate area)
point of inflection
logistic growth equation

27. When f '(x) is decreasing - f(x) is...
concave down
separate variables - integrate + C - use initial condition to find C - solve for y
if terms grow without bound - series diverges
(uv'-vu')/v²

28. Average Rate of Change
? f(x) dx on interval a to b = F(b) - F(a)
Slope of secant line between two points - use to estimate instantanous rate of change at a point.
positive
Limit as h approaches 0 of [f(a+h)-f(a)]/h

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

30. To find absolute maximum on closed interval [a - b] - you must consider...
Alternating series converges and general term diverges with another test
critical points and endpoints
Slope of tangent line at a point - value of derivative at a point
general term = 1/n^p - converges if p > 1

31. y = cos?¹(x) - y' =

32. Fundamental Theorem of Calculus
y' = sec²(x)
relative maximum
use trapezoids to evaluate integrals (estimate area)
? f(x) dx on interval a to b = F(b) - F(a)

33. Find interval of convergence
y' = -1/v(1 - x²)
if integral converges - series converges
use ratio test - set > 1 and solve absolute value equations - check endpoints
f '(g(x)) g'(x)

34. If f '(x) = 0 and f'(x) > 0 -...
product rule
use rectangles with right-endpoints to evaluate integrals (estimate area)
Slope of secant line between two points - use to estimate instantanous rate of change at a point.
f(x) has a relative minimum

35. Use partial fractions to integrate when...
uv - ? v du
corner - cusp - vertical tangent - discontinuity
integrand is a rational function with a factorable denominator
lim as n approaches 8 of ratio of (n+1) term/nth term > 1 - series converges

36. Given velocity vectors dx/dt and dy/dt - find total distance travelled
1/2 ? R² - r² over interval from a to b - find a & b by setting equations equal - solve for theta.
y' = -sin(x)
? v (dx/dt)² + (dy/dt)² over interval from a to b
negative

37. y = cot?¹(x) - y' =

38. Converges conditionally
product rule
substitution - parts - partial fractions
Alternating series converges and general term diverges with another test
f '(g(x)) g'(x)

39. area above x-axis is...
y' = -1/(1 + x²)
1/2 ? r² over interval from a to b - find a & b by setting r = 0 - solve for theta
corner - cusp - vertical tangent - discontinuity
positive

40. L'Hopitals rule
use rectangles with left-endpoints to evaluate integral (estimate area)
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 to find indeterminate limits - find derivative of numerator and denominator separately then evaluate limit
polynomial with infinite number of terms - includes general term

41. Taylor series
v(dx/dt)² + (dy/dt)² not an integral!
zero
? v (dx/dt)² + (dy/dt)² over interval from a to b
polynomial with infinite number of terms - includes general term

42. Converges absolutely
0/0 - 8/8 - 8*0 - 8 - 8 - 1^8 - 0° - 8°
Slope of tangent line at a point - value of derivative at a point
Alternating series converges and general term converges with another test
? v (dx/dt)² + (dy/dt)² over interval from a to b

43. When f '(x) changes from increasing to decreasing or decreasing to increasing - f(x) has a...
logistic growth equation
y' = 1/(1 + x²)
point of inflection
chain rule

44. Given v(t) find total distance travelled
increasing
? abs[v(t)] over interval a to b
y' = 1/v(1 - x²)
Limit as x approaches a of [f(x)-f(a)]/(x-a)

45. Alternate definition of derivative
logistic differential equation - M = carrying capacity
A function and it's derivative are in the integrand
zero
Limit as x approaches a of [f(x)-f(a)]/(x-a)

46. Product Rule

47. Find radius of convergence
? f(x) - g(x) over interval a to b - where f(x) is top function and g(x) is bottom function
use ratio test - set > 1 and solve absolute value equations - radius = center - endpoint
? A(x) dx over interval a to b - where A(x) is the area of the given cross-section in terms of x
two different types of functions are multiplied

48. 6th degree Taylor Polynomial
? A(x) dx over interval a to b - where A(x) is the area of the given cross-section in terms of x
uv' + vu'
positive
polynomial with finite number of terms - largest exponent is 6 - find all derivatives up to the 6th derivative

49. trapezoidal rule
positive
? A(x) dx over interval a to b - where A(x) is the area of the given cross-section in terms of x
Alternating series converges and general term converges with another test
use trapezoids to evaluate integrals (estimate area)

50. mean value theorem