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 increasing - f(x) is...
concave up
? f(x) dx on interval a to b = F(b) - F(a)
substitution - parts - partial fractions
y' = 1/v(1 - x²)


2. slope of horizontal line
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.
zero
e^x
f(x) has a relative maximum


3. Particle is moving to the right/up
velocity is positive
Limit as x approaches a of [f(x)-f(a)]/(x-a)
p ? r² dx over interval a to b - where r = distance from curve to axis of revolution
y' = sec(x)tan(x)


4. methods of integration
? f(x) dx integrate over interval a to b
Slope of tangent line at a point - value of derivative at a point
substitution - parts - partial fractions
f(x) has a relative minimum


5. Particle is moving to the left/down
use trapezoids to evaluate integrals (estimate area)
velocity is negative
f(x)
substitution - parts - partial fractions


6. p-series test
logistic growth equation
general term = 1/n^p - converges if p > 1
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
relative maximum


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

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8. Fundamental Theorem of Calculus
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)
Area of trapezoid
decreasing
? f(x) dx on interval a to b = F(b) - F(a)


9. absolute value of velocity
speed
y' = 1/v(1 - x²)
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)
general term = 1/n^p - converges if p > 1


10. When f '(x) is decreasing - f(x) is...
concave up
concave down
f(x) has a relative maximum
separate variables - integrate + C - use initial condition to find C - solve for y


11. Quotient Rule

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12. Find radius of convergence
positive
critical points and endpoints
y' = -csc²(x)
use ratio test - set > 1 and solve absolute value equations - radius = center - endpoint


13. y = e^x - y' = y' =
? f(x) - g(x) over interval a to b - where f(x) is top function and g(x) is bottom function
e^x
zero
negative


14. Volume of solid of revolution - washer
negative
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
y' = cos(x)
y' = sec(x)tan(x)


15. Eatio test
lim as n approaches 8 of ratio of (n+1) term/nth term > 1 - series converges
undefined
use tangent line to approximate values of the function
decreasing


16. average value of f(x)
1/(b-a) ? f(x) dx on interval a to b
y' = -sin(x)
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


17. When f '(x) is negative - f(x) is...
f(x) has a relative minimum
y' = -1/(1 + x²)
logistic growth equation
decreasing


18. Geometric series test
no limits - find antiderivative + C - use inital value to find C
? A(x) dx over interval a to b - where A(x) is the area of the given cross-section in terms of x
general term = a1r^n - converges if -1 < r < 1
derivative


19. Taylor series
Area of trapezoid
find first derivative - dy/dx = dy/dt / dx/dt - then find derivative of first derivative - then divide by dx/dt
y' = -csc²(x)
polynomial with infinite number of terms - includes general term


20. Converges absolutely
increasing
y' = -csc²(x)
Alternating series converges and general term converges with another test
y' = 1/(1 + x²)


21. 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.
use rectangles with right-endpoints to evaluate integrals (estimate area)
uv - ? v du
y' = 1/x


22. To find absolute maximum on closed interval [a - b] - you must consider...
a^x ln(a)
critical points and endpoints
Limit as x approaches a of [f(x)-f(a)]/(x-a)
concave up


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

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24. L'Hopitals rule
corner - cusp - vertical tangent - discontinuity
use to find indeterminate limits - find derivative of numerator and denominator separately then evaluate limit
use tangent line to approximate values of the function
concave up


25. area below x-axis is...
? v (dx/dt)² + (dy/dt)² over interval from a to b
negative
if terms grow without bound - series diverges
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)


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

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27. trapezoidal rule
? v (dx/dt)² + (dy/dt)² over interval from a to b
Alternating series converges and general term diverges with another test
concave up
use trapezoids to evaluate integrals (estimate area)


28. rate
0/0 - 8/8 - 8*0 - 8 - 8 - 1^8 - 0° - 8°
if terms grow without bound - series diverges
derivative
use trapezoids to evaluate integrals (estimate area)


29. Converges conditionally
use ratio test - set > 1 and solve absolute value equations - radius = center - endpoint
Alternating series converges and general term diverges with another test
1/2 ? r² over interval from a to b - find a & b by setting r = 0 - solve for theta
logistic growth equation


30. If f '(x) = 0 and f'(x) > 0 -...
increasing
zero
f(x) has a relative minimum
if lim as n approaches 8 of ratio of comparison series/general term is positive and finite - then series behaves like comparison series


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

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32. definite integral
use ratio test - set > 1 and solve absolute value equations - check endpoints
y' = -1/(1 + x²)
y' = -csc²(x)
has limits a & b - find antiderivative - F(b) - F(a)


33. Volume of solid with base in the plane and given cross-section
general term = 1/n^p - converges if p > 1
lim as n approaches 8 of ratio of (n+1) term/nth term > 1 - series converges
? A(x) dx over interval a to b - where A(x) is the area of the given cross-section in terms of x
point of inflection


34. y = cos²(3x)
chain rule
(uv'-vu')/v²
draw short segments representing slope at each point
? v(1 + (dy/dx)²) dx over interval a to b


35. Area between two curves
if integral converges - series converges
? f(x) - g(x) over interval a to b - where f(x) is top function and g(x) is bottom function
velocity is positive
concave up


36. indefinite integral
no limits - find antiderivative + C - use inital value to find C
y' = -sin(x)
relative maximum
use rectangles with left-endpoints to evaluate integral (estimate area)


37. use integration by parts when...
logistic differential equation - M = carrying capacity
product rule
use ratio test - set > 1 and solve absolute value equations - radius = center - endpoint
two different types of functions are multiplied


38. area above x-axis is...
if terms grow without bound - series diverges
velocity is positive
p ? r² dx over interval a to b - where r = distance from curve to axis of revolution
positive


39. When f '(x) changes from negative to positive - f(x) has a...
y' = -sin(x)
use tangent line to approximate values of the function
relative minimum
p ? r² dx over interval a to b - where r = distance from curve to axis of revolution


40. When f '(x) changes from increasing to decreasing or decreasing to increasing - f(x) has a...
use trapezoids to evaluate integrals (estimate area)
point of inflection
speed
? v (dx/dt)² + (dy/dt)² over interval from a to b


41. dP/dt = kP(M - P)
polynomial with finite number of terms - largest exponent is 6 - find all derivatives up to the 6th derivative
logistic differential equation - M = carrying capacity
? f(x) - g(x) over interval a to b - where f(x) is top function and g(x) is bottom function
product rule


42. y = x cos(x) - state rule used to find derivative
draw short segments representing slope at each point
f(x) has a relative minimum
? v(t) over interval a to b
product rule


43. Use partial fractions to integrate when...
lim as n approaches 8 of ratio of (n+1) term/nth term > 1 - series converges
integrand is a rational function with a factorable denominator
1/2 ? r² over interval from a to b - find a & b by setting r = 0 - solve for theta
y' = -sin(x)


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

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45. Integral test
if integral converges - series converges
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
positive
Limit as x approaches a of [f(x)-f(a)]/(x-a)


46. y = tan?¹(x) - y' =

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47. [(h1 - h2)/2]*base
p ? r² dx over interval a to b - where r = distance from curve to axis of revolution
Area of trapezoid
use ratio test - set > 1 and solve absolute value equations - check endpoints
(uv'-vu')/v²


48. Given velocity vectors dx/dt and dy/dt - find total distance travelled
? v(t) over interval a to b
? v (dx/dt)² + (dy/dt)² over interval from a to b
concave down
f '(g(x)) g'(x)


49. Alternating series tes
y' = -csc²(x)
? f(x) dx on interval a to b = F(b) - F(a)
use rectangles with left-endpoints to evaluate integral (estimate area)
lim as n approaches zero of general term = 0 and terms decrease - series converges


50. y = cos?¹(x) - y' =

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