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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. average value of f(x)
y' = -1/(1 + x²)
y' = -sin(x)
1/(b-a) ? f(x) dx on interval a to b
? v (dx/dt)² + (dy/dt)² over interval from a to b

2. Alternate definition of derivative
y' = 1/v(1 - x²)
Limit as x approaches a of [f(x)-f(a)]/(x-a)
integrand is a rational function with a factorable denominator
a^x ln(a)

3. rate
concave down
a^x ln(a)
general term = a1r^n - converges if -1 < r < 1
derivative

4. y = cos?¹(x) - y' =

5. Product Rule

6. Volume of solid of revolution - no washer
? v(t) over interval a to b
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.
p ? r² dx over interval a to b - where r = distance from curve to axis of revolution
polynomial with finite number of terms - largest exponent is 6 - find all derivatives up to the 6th derivative

7. To draw a slope field - plug (x -y) coordinates into differential equation...
draw short segments representing slope at each point
y' = -csc(x)cot(x)
? v(1 + (dy/dx)²) dx over interval a to b
Alternating series converges and general term diverges with another test

8. When f '(x) is negative - f(x) is...
1/2 ? r² over interval from a to b - find a & b by setting r = 0 - solve for theta
uv - ? v du
decreasing
f(x) has a relative minimum

9. Chain Rule

10. trapezoidal rule
uv' + vu'
y' = cos(x)
velocity is negative
use trapezoids to evaluate integrals (estimate area)

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

12. Particle is moving to the right/up
velocity is positive
y' = 1/(1 + x²)
find first derivative - dy/dx = dy/dt / dx/dt - then find derivative of first derivative - then divide by dx/dt
? f(x) dx on interval a to b = F(b) - F(a)

13. Volume of solid of revolution - washer
y' = sec²(x)
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
? v (dx/dt)² + (dy/dt)² over interval from a to b

14. nth term test
y' = -sin(x)
zero
if terms grow without bound - series diverges
concave down

15. Find radius of convergence
use ratio test - set > 1 and solve absolute value equations - radius = center - endpoint
integrand is a rational function with a factorable denominator
velocity is positive
1/(b-a) ? f(x) dx on interval a to b

16. y = cot?¹(x) - y' =

17. 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
concave down
uv' + vu'
? f(x) - g(x) over interval a to b - where f(x) is top function and g(x) is bottom function

18. mean value theorem

19. Given v(t) find total distance travelled
Limit as h approaches 0 of [f(a+h)-f(a)]/h
? abs[v(t)] over interval a to b
? v (dx/dt)² + (dy/dt)² over interval from a to b
f(x)

20. absolute value of velocity
find first derivative - dy/dx = dy/dt / dx/dt - then find derivative of first derivative - then divide by dx/dt
integrand is a rational function with a factorable denominator
speed
use trapezoids to evaluate integrals (estimate area)

21. right riemann sum
use rectangles with right-endpoints to evaluate integrals (estimate area)
v(dx/dt)² + (dy/dt)² not an integral!
speed
positive

22. When f '(x) is positive - f(x) is...
if integral converges - series converges
use ratio test - set > 1 and solve absolute value equations - check endpoints
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
increasing

23. y = e^x - y' = y' =
no limits - find antiderivative + C - use inital value to find C
Alternating series converges and general term converges with another test
speed
e^x

24. Instantenous Rate of Change
? abs[v(t)] over interval a to b
(uv'-vu')/v²
Slope of tangent line at a point - value of derivative at a point
substitution - parts - partial fractions

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

26. slope of vertical line
0/0 - 8/8 - 8*0 - 8 - 8 - 1^8 - 0° - 8°
(uv'-vu')/v²
increasing
undefined

27. slope of horizontal line
y' = -1/(1 + x²)
zero
f(x)
velocity is negative

28. Intermediate Value Theorem
use rectangles with left-endpoints to evaluate integral (estimate area)
? abs[v(t)] over interval a to b
polynomial with infinite number of terms - includes general term
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.

29. To find particular solution to differential equation - dy/dx = x/y...
polynomial with infinite number of terms - includes general term
polynomial with finite number of terms - largest exponent is 6 - find all derivatives up to the 6th derivative
use trapezoids to evaluate integrals (estimate area)
separate variables - integrate + C - use initial condition to find C - solve for y

30. ? u dv =
? f(x) - g(x) over interval a to b - where f(x) is top function and g(x) is bottom function
uv - ? v du
product rule
relative minimum

31. Area inside one polar curve and outside another polar curve
use trapezoids to evaluate integrals (estimate area)
y' = -sin(x)
1/2 ? R² - r² over interval from a to b - find a & b by setting equations equal - solve for theta.
A function and it's derivative are in the integrand

32. P = M / (1 + Ae^(-Mkt))
positive
draw short segments representing slope at each point
logistic growth equation
polynomial with infinite number of terms - includes general term

33. area below x-axis is...
y' = sec(x)tan(x)
y' = cos(x)
negative
general term = 1/n^p - converges if p > 1

34. 6th degree Taylor Polynomial
? v (dx/dt)² + (dy/dt)² over interval from a to b
draw short segments representing slope at each point
polynomial with finite number of terms - largest exponent is 6 - find all derivatives up to the 6th derivative
critical points and endpoints

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

36. When f '(x) is decreasing - f(x) is...
if integral converges - series converges
y' = cos(x)
concave up
concave down

37. y = cos²(3x)
? v (dx/dt)² + (dy/dt)² over interval from a to b
chain rule
Alternating series converges and general term converges with another test
increasing

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

39. When f '(x) changes from increasing to decreasing or decreasing to increasing - f(x) has a...
derivative
point of inflection
two different types of functions are multiplied
1/(b-a) ? f(x) dx on interval a to b

40. p-series test
velocity is positive
general term = 1/n^p - converges if p > 1
lim as n approaches zero of general term = 0 and terms decrease - series converges
f(x) has a relative maximum

41. y = x cos(x) - state rule used to find derivative
concave down
draw short segments representing slope at each point
y' = -1/v(1 - x²)
product rule

42. y = sin?¹(x) - y' =

43. Average Rate of Change
v(dx/dt)² + (dy/dt)² not an integral!
velocity is positive
Slope of secant line between two points - use to estimate instantanous rate of change at a point.
1/2 ? r² over interval from a to b - find a & b by setting r = 0 - solve for theta

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

45. Area between two curves
use trapezoids to evaluate integrals (estimate area)
? f(x) - g(x) over interval a to b - where f(x) is top function and g(x) is bottom function
? f(x) dx on interval a to b = F(b) - F(a)
0/0 - 8/8 - 8*0 - 8 - 8 - 1^8 - 0° - 8°

46. Limit comparison test
f(x) has a relative minimum
? f(x) - g(x) over interval a to b - where f(x) is top function and g(x) is bottom function
1/(b-a) ? f(x) dx on interval a to b
if lim as n approaches 8 of ratio of comparison series/general term is positive and finite - then series behaves like comparison series

47. Linearization
two different types of functions are multiplied
f(x) has a relative maximum
v(dx/dt)² + (dy/dt)² not an integral!
use tangent line to approximate values of the function

48. area under a curve
lim as n approaches 8 of ratio of (n+1) term/nth term > 1 - series converges
1/(b-a) ? f(x) dx on interval a to b
? f(x) dx integrate over interval a to b
chain rule

49. Particle is moving to the left/down
relative minimum
velocity is negative
use tangent line to approximate values of the function
if terms grow without bound - series diverges

50. Given velocity vectors dx/dt and dy/dt - find speed
undefined
general term = a1r^n - converges if -1 < r < 1
speed
v(dx/dt)² + (dy/dt)² not an integral!