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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. When is a function not differentiable
f '(g(x)) g'(x)
use tangent line to approximate values of the function
Area of trapezoid
corner - cusp - vertical tangent - discontinuity

2. mean value theorem

3. y = x cos(x) - state rule used to find derivative
e^x
product rule
integrand is a rational function with a factorable denominator
derivative

4. right riemann sum
Alternating series converges and general term converges with another test
use rectangles with right-endpoints to evaluate integrals (estimate area)
relative maximum
speed

5. Geometric series test
general term = a1r^n - converges if -1 < r < 1
? v (dx/dt)² + (dy/dt)² over interval from a to b
if integral converges - series converges
e^x

6. ? u dv =
logistic growth equation
y' = -csc(x)cot(x)
uv - ? v du
Limit as h approaches 0 of [f(a+h)-f(a)]/h

7. Particle is moving to the left/down
e^x
zero
product rule
velocity is negative

8. Chain Rule

9. Area inside polar curve
1/2 ? r² over interval from a to b - find a & b by setting r = 0 - solve for theta
has limits a & b - find antiderivative - F(b) - F(a)
logistic differential equation - M = carrying capacity
y' = 1/(1 + x²)

10. Formal definition of derivative
no limits - find antiderivative + C - use inital value to find C
chain rule
Limit as h approaches 0 of [f(a+h)-f(a)]/h
find first derivative - dy/dx = dy/dt / dx/dt - then find derivative of first derivative - then divide by dx/dt

11. Alternate definition of derivative
Limit as x approaches a of [f(x)-f(a)]/(x-a)
concave up
uv' + vu'
zero

12. Given v(t) find total distance travelled
1/(b-a) ? f(x) dx on interval a to b
Alternating series converges and general term diverges with another test
critical points and endpoints
? abs[v(t)] over interval a to b

13. [(h1 - h2)/2]*base
use tangent line to approximate values of the function
Slope of secant line between two points - use to estimate instantanous rate of change at a point.
Area of trapezoid
find first derivative - dy/dx = dy/dt / dx/dt - then find derivative of first derivative - then divide by dx/dt

14. Linearization
y' = 1/v(1 - x²)
use tangent line to approximate values of the function
polynomial with finite number of terms - largest exponent is 6 - find all derivatives up to the 6th derivative
polynomial with infinite number of terms - includes general term

15. Second derivative of parametrically defined curve
concave up
zero
find first derivative - dy/dx = dy/dt / dx/dt - then find derivative of first derivative - then divide by dx/dt
v(dx/dt)² + (dy/dt)² not an integral!

16. area under a curve
p ? r² dx over interval a to b - where r = distance from curve to axis of revolution
uv - ? v du
negative
? f(x) dx integrate over interval a to b

17. Limit comparison test
velocity is positive
use ratio test - set > 1 and solve absolute value equations - radius = center - endpoint
concave up
if lim as n approaches 8 of ratio of comparison series/general term is positive and finite - then series behaves like comparison series

18. Length of parametric curve
concave down
find first derivative - dy/dx = dy/dt / dx/dt - then find derivative of first derivative - then divide by dx/dt
? v (dx/dt)² + (dy/dt)² over interval from a to b
? v(1 + (dy/dx)²) dx over interval a to b

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

20. absolute value of velocity
speed
corner - cusp - vertical tangent - discontinuity
use ratio test - set > 1 and solve absolute value equations - check endpoints
concave down

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

22. When f '(x) is positive - f(x) is...
increasing
concave up
point of inflection
A function and it's derivative are in the integrand

23. trapezoidal rule
find first derivative - dy/dx = dy/dt / dx/dt - then find derivative of first derivative - then divide by dx/dt
use trapezoids to evaluate integrals (estimate area)
polynomial with finite number of terms - largest exponent is 6 - find all derivatives up to the 6th derivative
use ratio test - set > 1 and solve absolute value equations - radius = center - endpoint

24. When f '(x) is increasing - f(x) is...
if terms grow without bound - series diverges
concave up
y' = 1/v(1 - x²)
use ratio test - set > 1 and solve absolute value equations - check endpoints

25. 6th degree Taylor Polynomial
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
corner - cusp - vertical tangent - discontinuity
polynomial with finite number of terms - largest exponent is 6 - find all derivatives up to the 6th derivative

26. Alternating series tes
e^x
f(x) has a relative minimum
lim as n approaches zero of general term = 0 and terms decrease - series converges
if lim as n approaches 8 of ratio of comparison series/general term is positive and finite - then series behaves like comparison series

27. Given v(t) find displacement
1/2 ? R² - r² over interval from a to b - find a & b by setting equations equal - solve for theta.
y' = 1/(1 + x²)
? v(t) over interval a to b
v(dx/dt)² + (dy/dt)² not an integral!

28. Volume of solid of revolution - washer
0/0 - 8/8 - 8*0 - 8 - 8 - 1^8 - 0° - 8°
polynomial with finite number of terms - largest exponent is 6 - find all derivatives up to the 6th derivative
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 tangent line to approximate values of the function

29. When f '(x) changes from increasing to decreasing or decreasing to increasing - f(x) has a...
point of inflection
Limit as x approaches a of [f(x)-f(a)]/(x-a)
use tangent line to approximate values of the function
velocity is negative

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

31. Volume of solid of revolution - no washer
draw short segments representing slope at each point
use ratio test - set > 1 and solve absolute value equations - radius = center - endpoint
y' = 1/(x lna)
p ? r² dx over interval a to b - where r = distance from curve to axis of revolution

32. Area inside one polar curve and outside another polar curve
velocity is negative
find first derivative - dy/dx = dy/dt / dx/dt - then find derivative of first derivative - then divide by dx/dt
1/2 ? R² - r² over interval from a to b - find a & b by setting equations equal - solve for theta.
use tangent line to approximate values of the function

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

34. Particle is moving to the right/up
Slope of tangent line at a point - value of derivative at a point
positive
e^x
velocity is positive

35. Average Rate of Change
use tangent line to approximate values of the function
Slope of secant line between two points - use to estimate instantanous rate of change at a point.
use ratio test - set > 1 and solve absolute value equations - radius = center - endpoint
separate variables - integrate + C - use initial condition to find C - solve for y

36. y = e^x - y' = y' =
y' = -csc(x)cot(x)
e^x
use rectangles with right-endpoints to evaluate integrals (estimate area)
A function and it's derivative are in the integrand

37. indefinite integral
chain rule
has limits a & b - find antiderivative - F(b) - F(a)
no limits - find antiderivative + C - use inital value to find C
use ratio test - set > 1 and solve absolute value equations - radius = center - endpoint

38. average value of f(x)
1/(b-a) ? f(x) dx on 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
no limits - find antiderivative + C - use inital value to find C
logistic differential equation - M = carrying capacity

39. rate
derivative
lim as n approaches 8 of ratio of (n+1) term/nth term > 1 - series converges
Limit as h approaches 0 of [f(a+h)-f(a)]/h
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)

40. area below x-axis is...
negative
y' = -csc²(x)
velocity is negative
concave up

41. Given velocity vectors dx/dt and dy/dt - find speed
polynomial with finite number of terms - largest exponent is 6 - find all derivatives up to the 6th derivative
integrand is a rational function with a factorable denominator
v(dx/dt)² + (dy/dt)² not an integral!
find first derivative - dy/dx = dy/dt / dx/dt - then find derivative of first derivative - then divide by dx/dt

42. methods of integration
if lim as n approaches 8 of ratio of comparison series/general term is positive and finite - then series behaves like comparison series
speed
substitution - parts - partial fractions
f(x) has a relative minimum

43. If f '(x) = 0 and f'(x) < 0 -...
f(x) has a relative maximum
y' = -1/(1 + x²)
Limit as x approaches a of [f(x)-f(a)]/(x-a)
point of inflection

44. Instantenous Rate of Change
y' = -sin(x)
Slope of tangent line at a point - value of derivative at a point
lim as n approaches zero of general term = 0 and terms decrease - series converges
y' = 1/(1 + x²)

45. Find interval of convergence
use ratio test - set > 1 and solve absolute value equations - check endpoints
use ratio test - set > 1 and solve absolute value equations - radius = center - endpoint
Limit as x approaches a of [f(x)-f(a)]/(x-a)
no limits - find antiderivative + C - use inital value to find C

46. To draw a slope field - plug (x -y) coordinates into differential equation...
y' = -csc(x)cot(x)
zero
if integral converges - series converges
draw short segments representing slope at each point

47. slope of vertical line
f(x) has a relative maximum
speed
undefined
a^x ln(a)

48. y = a^x - y' = y' =
Alternating series converges and general term converges with another test
a^x ln(a)
general term = a1r^n - converges if -1 < r < 1
polynomial with finite number of terms - largest exponent is 6 - find all derivatives up to the 6th derivative

49. Intermediate Value Theorem
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.
y' = -csc(x)cot(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.

50. Converges absolutely
Alternating series converges and general term converges with another test
if lim as n approaches 8 of ratio of comparison series/general term is positive and finite - then series behaves like comparison series
0/0 - 8/8 - 8*0 - 8 - 8 - 1^8 - 0° - 8°
Area of trapezoid