AP Calculus Bc

Instructions:

• Answer 50 questions in 15 minutes.
• If you are not ready to take this test, you can study here.
• Match each statement with the correct term.
• Don't refresh. All questions and answers are randomly picked and ordered every time you load a test.

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 f '(x) is positive - f(x) is...
decreasing
increasing
use trapezoids to evaluate integrals (estimate area)
y' = -csc²(x)


2. Area inside polar curve
1/2 ? r² over interval from a to b - find a & b by setting r = 0 - solve for theta
velocity is positive
product rule
f(x) has a relative minimum


3. y = a^x - y' = y' =
y' = -1/(1 + x²)
a^x ln(a)
y' = 1/(x lna)
velocity is positive


4. When f '(x) changes from increasing to decreasing or decreasing to increasing - f(x) has a...
Slope of tangent line at a point - value of derivative at a point
point of inflection
e^x
1/2 ? R² - r² over interval from a to b - find a & b by setting equations equal - solve for theta.


5. use substitution to integrate when

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6. Alternate definition of derivative
Limit as h approaches 0 of [f(a+h)-f(a)]/h
increasing
velocity is positive
Limit as x approaches a of [f(x)-f(a)]/(x-a)


7. Integral test
product rule
(uv'-vu')/v²
find first derivative - dy/dx = dy/dt / dx/dt - then find derivative of first derivative - then divide by dx/dt
if integral converges - series converges


8. y = cos?¹(x) - y' =

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9. y = sin?¹(x) - y' =

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10. Fundamental Theorem of Calculus
lim as n approaches 8 of ratio of (n+1) term/nth term > 1 - series converges
? f(x) dx on interval a to b = F(b) - F(a)
corner - cusp - vertical tangent - discontinuity
? f(x) - g(x) over interval a to b - where f(x) is top function and g(x) is bottom function


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

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12. nth term test
if terms grow without bound - series diverges
y' = -sin(x)
use tangent line to approximate values of the function
use ratio test - set > 1 and solve absolute value equations - check endpoints


13. Product Rule

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14. Particle is moving to the left/down
velocity is negative
use ratio test - set > 1 and solve absolute value equations - radius = center - endpoint
critical points and endpoints
use rectangles with left-endpoints to evaluate integral (estimate area)


15. Use partial fractions to integrate when...
if integral converges - series converges
integrand is a rational function with a factorable denominator
? v(t) over interval a to b
y' = -csc(x)cot(x)


16. L'Hopitals rule
use to find indeterminate limits - find derivative of numerator and denominator separately then evaluate limit
y' = cos(x)
uv' + vu'
y' = -1/v(1 - x²)


17. Quotient Rule

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18. dP/dt = kP(M - P)
draw short segments representing slope at each point
y' = -1/(1 + x²)
? f(x) dx integrate over interval a to b
logistic differential equation - M = carrying capacity


19. p-series test
general term = 1/n^p - converges if p > 1
Limit as h approaches 0 of [f(a+h)-f(a)]/h
? v (dx/dt)² + (dy/dt)² over interval from a to b
relative minimum


20. y = tan?¹(x) - y' =

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21. To draw a slope field - plug (x -y) coordinates into differential equation...
quotient rule
draw short segments representing slope at each point
negative
y' = sec(x)tan(x)


22. y = cos²(3x)
y' = -1/(1 + x²)
point of inflection
chain rule
negative


23. When f '(x) changes fro positive to negative - f(x) has a...
Alternating series converges and general term converges with another test
relative maximum
Slope of secant line between two points - use to estimate instantanous rate of change at a point.
corner - cusp - vertical tangent - discontinuity


24. area below x-axis is...
velocity is negative
y' = 1/(1 + x²)
y' = 1/(x lna)
negative


25. Instantenous Rate of Change
e^x
? f(x) - g(x) over interval a to b - where f(x) is top function and g(x) is bottom function
concave down
Slope of tangent line at a point - value of derivative at a point


26. average value of f(x)
use rectangles with right-endpoints to evaluate integrals (estimate area)
speed
1/(b-a) ? f(x) dx on interval a to b
0/0 - 8/8 - 8*0 - 8 - 8 - 1^8 - 0° - 8°


27. Length of parametric curve
y' = -csc(x)cot(x)
f(x) has a relative minimum
? v (dx/dt)² + (dy/dt)² over interval from 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.


28. indefinite integral
two different types of functions are multiplied
derivative
no limits - find antiderivative + C - use inital value to find C
? v(t) over interval a to b


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


30. Find interval of convergence
Alternating series converges and general term diverges with another test
use trapezoids to evaluate integrals (estimate area)
use ratio test - set > 1 and solve absolute value equations - check endpoints
use tangent line to approximate values of the function


31. Intermediate Value Theorem
p ? r² dx over interval a to b - where r = distance from curve to axis of revolution
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.
increasing
? f(x) dx on interval a to b = F(b) - F(a)


32. Area inside one polar curve and outside another polar curve
1/2 ? R² - r² over interval from a to b - find a & b by setting equations equal - solve for theta.
use ratio test - set > 1 and solve absolute value equations - check endpoints
Limit as h approaches 0 of [f(a+h)-f(a)]/h
Alternating series converges and general term converges with another test


33. slope of vertical line
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)
relative minimum
use rectangles with left-endpoints to evaluate integral (estimate area)
undefined


34. right riemann sum
integrand is a rational function with a factorable denominator
use rectangles with right-endpoints to evaluate integrals (estimate area)
v(dx/dt)² + (dy/dt)² not an integral!
y' = 1/(1 + x²)


35. To find absolute maximum on closed interval [a - b] - you must consider...
general term = 1/n^p - converges if p > 1
use to find indeterminate limits - find derivative of numerator and denominator separately then evaluate limit
critical points and endpoints
a^x ln(a)


36. Length of curve
logistic differential equation - M = carrying capacity
concave up
? v(1 + (dy/dx)²) dx over interval a to b
? v (dx/dt)² + (dy/dt)² over interval from a to b


37. use integration by parts when...
Area of trapezoid
two different types of functions are multiplied
use ratio test - set > 1 and solve absolute value equations - check endpoints
polynomial with finite number of terms - largest exponent is 6 - find all derivatives up to the 6th derivative


38. slope of horizontal line
decreasing
f(x) has a relative minimum
zero
velocity is positive


39. Taylor series
polynomial with infinite number of terms - includes general term
e^x
(uv'-vu')/v²
y' = -csc²(x)


40. Chain Rule

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41. If f '(x) = 0 and f'(x) > 0 -...
f(x) has a relative maximum
two different types of functions are multiplied
f(x) has a relative minimum
uv - ? v du


42. left riemann sum
use rectangles with left-endpoints to evaluate integral (estimate area)
zero
use ratio test - set > 1 and solve absolute value equations - check endpoints
? v(t) over interval a to b


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


44. When f '(x) is increasing - f(x) is...
derivative
if lim as n approaches 8 of ratio of comparison series/general term is positive and finite - then series behaves like comparison series
concave up
critical points and endpoints


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

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46. absolute value of velocity
speed
use rectangles with right-endpoints to evaluate integrals (estimate area)
1/2 ? R² - r² over interval from a to b - find a & b by setting equations equal - solve for theta.
general term = a1r^n - converges if -1 < r < 1


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

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48. Alternating series tes
lim as n approaches zero of general term = 0 and terms decrease - series converges
? v(t) over interval a to b
no limits - find antiderivative + C - use inital value to find C
if integral converges - series converges


49. When f '(x) is decreasing - f(x) is...
logistic differential equation - M = carrying capacity
? v (dx/dt)² + (dy/dt)² over interval from a to b
concave down
use to find indeterminate limits - find derivative of numerator and denominator separately then evaluate limit


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

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