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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 = cot?¹(x) - y' =

2. Use partial fractions to integrate when...
integrand is a rational function with a factorable denominator
y' = cos(x)
find first derivative - dy/dx = dy/dt / dx/dt - then find derivative of first derivative - then divide by dx/dt
concave up

3. y = sin?¹(x) - y' =

4. To find absolute maximum on closed interval [a - b] - you must consider...
critical points and endpoints
concave down
y' = sec(x)tan(x)
? v(1 + (dy/dx)²) dx over interval a to b

5. Instantenous Rate of Change
A function and it's derivative are in the integrand
y' = 1/(1 + x²)
Slope of tangent line at a point - value of derivative at a point
y' = -sin(x)

6. Volume of solid of revolution - washer
if lim as n approaches 8 of ratio of comparison series/general term is positive and finite - then series behaves like comparison series
1/2 ? r² over interval from a to b - find a & b by setting r = 0 - solve for theta
y' = -sin(x)
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

7. Length of curve
speed
separate variables - integrate + C - use initial condition to find C - solve for y
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(1 + (dy/dx)²) dx over interval a to b

8. average value of f(x)
1/(b-a) ? f(x) dx on interval a to b
logistic growth equation
y' = 1/(x lna)
polynomial with finite number of terms - largest exponent is 6 - find all derivatives up to the 6th derivative

9. Volume of solid of revolution - no washer
? v(t) over interval a to b
p ? r² dx over interval a to b - where r = distance from curve to axis of revolution
use trapezoids to evaluate integrals (estimate area)
draw short segments representing slope at each point

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

11. Converges conditionally
a^x ln(a)
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.
Alternating series converges and general term diverges with another test
1/2 ? R² - r² over interval from a to b - find a & b by setting equations equal - solve for theta.

12. Linearization
use tangent line to approximate values of the function
zero
f '(g(x)) g'(x)
uv - ? v du

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

14. slope of horizontal line
zero
? v (dx/dt)² + (dy/dt)² over interval from a to b
Limit as x approaches a of [f(x)-f(a)]/(x-a)
decreasing

15. L'Hopitals rule
use to find indeterminate limits - find derivative of numerator and denominator separately then evaluate limit
Limit as h approaches 0 of [f(a+h)-f(a)]/h
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!

16. right riemann sum
v(dx/dt)² + (dy/dt)² not an integral!
use rectangles with right-endpoints to evaluate integrals (estimate area)
? v(1 + (dy/dx)²) dx over interval a to b
use tangent line to approximate values of the function

17. left riemann sum
substitution - parts - partial fractions
use rectangles with left-endpoints to evaluate integral (estimate area)
uv - ? v du
use ratio test - set > 1 and solve absolute value equations - check endpoints

18. Intermediate Value Theorem
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.
Slope of tangent line at a point - value of derivative at a point
increasing
corner - cusp - vertical tangent - discontinuity

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

20. Given velocity vectors dx/dt and dy/dt - find speed
integrand is a rational function with a factorable denominator
v(dx/dt)² + (dy/dt)² not an integral!
Slope of secant line between two points - use to estimate instantanous rate of change at a point.
increasing

21. Given velocity vectors dx/dt and dy/dt - find total distance travelled
? v (dx/dt)² + (dy/dt)² over interval from 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
general term = a1r^n - converges if -1 < r < 1
concave up

22. Alternating series tes
a^x ln(a)
f(x) has a relative maximum
lim as n approaches zero of general term = 0 and terms decrease - series converges
y' = -csc²(x)

23. Find interval of convergence
chain rule
use ratio test - set > 1 and solve absolute value equations - check endpoints
1/(b-a) ? f(x) dx on interval a to b
0/0 - 8/8 - 8*0 - 8 - 8 - 1^8 - 0° - 8°

24. When is a function not differentiable
use trapezoids to evaluate integrals (estimate area)
use ratio test - set > 1 and solve absolute value equations - radius = center - endpoint
relative minimum
corner - cusp - vertical tangent - discontinuity

25. Given v(t) find total distance travelled
f '(g(x)) g'(x)
Slope of tangent line at a point - value of derivative at a point
? abs[v(t)] over interval a to b
Slope of secant line between two points - use to estimate instantanous rate of change at a point.

26. If f '(x) = 0 and f'(x) < 0 -...
no limits - find antiderivative + C - use inital value to find C
f(x) has a relative maximum
general term = a1r^n - converges if -1 < r < 1
polynomial with infinite number of terms - includes general term

27. [(h1 - h2)/2]*base
y' = 1/(x lna)
Area of trapezoid
corner - cusp - vertical tangent - discontinuity
use trapezoids to evaluate integrals (estimate area)

28. Given v(t) find displacement
y' = sec(x)tan(x)
1/2 ? r² over interval from a to b - find a & b by setting r = 0 - solve for theta
? v(t) over interval a to b
relative minimum

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

30. y = cos?¹(x) - y' =

31. definite integral
uv' + vu'
? f(x) - g(x) over interval a to b - where f(x) is top function and g(x) is bottom function
use tangent line to approximate values of the function
has limits a & b - find antiderivative - F(b) - F(a)

32. Find radius of convergence
two different types of functions are multiplied
use ratio test - set > 1 and solve absolute value equations - radius = center - endpoint
point of inflection
relative maximum

33. Particle is moving to the right/up
? v(t) over interval a to b
Limit as x approaches a of [f(x)-f(a)]/(x-a)
corner - cusp - vertical tangent - discontinuity
velocity is positive

34. When f '(x) is increasing - f(x) is...
? f(x) - g(x) over interval a to b - where f(x) is top function and g(x) is bottom function
concave up
zero
polynomial with infinite number of terms - includes general term

35. Eatio test
f(x)
? f(x) dx on interval a to b = F(b) - F(a)
speed
lim as n approaches 8 of ratio of (n+1) term/nth term > 1 - series converges

36. Average Rate of Change
uv - ? v du
Slope of secant line between two points - use to estimate instantanous rate of change at a point.
corner - cusp - vertical tangent - discontinuity
Alternating series converges and general term converges with another test

37. area under a curve
Slope of tangent line at a point - value of derivative at a point
? f(x) dx integrate over interval a to b
? v(1 + (dy/dx)²) dx over interval a to b
separate variables - integrate + C - use initial condition to find C - solve for y

38. Formal definition of derivative
no limits - find antiderivative + C - use inital value to find C
Slope of secant line between two points - use to estimate instantanous rate of change at a point.
Limit as h approaches 0 of [f(a+h)-f(a)]/h
two different types of functions are multiplied

39. trapezoidal rule
y' = 1/x
use trapezoids to evaluate integrals (estimate area)
no limits - find antiderivative + C - use inital value to find C
1/(b-a) ? f(x) dx on interval a to b

40. y = tan?¹(x) - y' =

41. P = M / (1 + Ae^(-Mkt))
f(x)
logistic growth equation
draw short segments representing slope at each point
if integral converges - series converges

42. If f '(x) = 0 and f'(x) > 0 -...
? v (dx/dt)² + (dy/dt)² over interval from a to b
f(x) has a relative minimum
draw short segments representing slope at each point
uv' + vu'

43. Chain Rule

44. Area between two curves
lim as n approaches zero of general term = 0 and terms decrease - series converges
f(x) has a relative maximum
? f(x) - g(x) over interval a to b - where f(x) is top function and g(x) is bottom function
velocity is negative

45. Area inside polar curve
Area of trapezoid
draw short segments representing slope at each point
concave up
1/2 ? r² over interval from a to b - find a & b by setting r = 0 - solve for theta

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

47. Quotient Rule

48. Volume of solid with base in the plane and given cross-section
(uv'-vu')/v²
? A(x) dx over interval a to b - where A(x) is the area of the given cross-section in terms of x
speed
relative maximum

49. When f '(x) is decreasing - f(x) is...
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 rectangles with left-endpoints to evaluate integral (estimate area)
concave down
use trapezoids to evaluate integrals (estimate area)

50. methods of integration
? v(1 + (dy/dx)²) dx over interval a to b
substitution - parts - partial fractions
chain rule
use ratio test - set > 1 and solve absolute value equations - check endpoints