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. mean value theorem

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2. Particle is moving to the right/up
Area of trapezoid
velocity is positive
relative maximum
1/(b-a) ? f(x) dx on interval a to b


3. Chain Rule

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4. Alternate definition of derivative
Limit as x approaches a of [f(x)-f(a)]/(x-a)
has limits a & b - find antiderivative - F(b) - F(a)
separate variables - integrate + C - use initial condition to find C - solve for y
f(x)


5. 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 maximum
? f(x) dx integrate over interval a to b
undefined


6. To find particular solution to differential equation - dy/dx = x/y...
positive
y' = -sin(x)
separate variables - integrate + C - use initial condition to find C - solve for y
1/2 ? r² over interval from a to b - find a & b by setting r = 0 - solve for theta


7. If g(x) = ? f(t) dt on interval 2 to x - then g'(x) =...
Limit as x approaches a of [f(x)-f(a)]/(x-a)
f(x)
Limit as h approaches 0 of [f(a+h)-f(a)]/h
y' = 1/(x lna)


8. area above x-axis is...
y' = 1/(x lna)
velocity is positive
y' = -sin(x)
positive


9. y = e^x - y' = y' =
lim as n approaches 8 of ratio of (n+1) term/nth term > 1 - series converges
logistic growth equation
? v (dx/dt)² + (dy/dt)² over interval from a to b
e^x


10. Geometric series test
general term = a1r^n - converges if -1 < r < 1
uv' + vu'
decreasing
e^x


11. Volume of solid of revolution - washer
? v(1 + (dy/dx)²) dx over interval a to b
two different types of functions are multiplied
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)
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


12. When f '(x) is decreasing - f(x) is...
zero
two different types of functions are multiplied
y' = -csc²(x)
concave down


13. When f '(x) changes from increasing to decreasing or decreasing to increasing - f(x) has a...
point of inflection
Alternating series converges and general term diverges with another test
? A(x) dx over interval a to b - where A(x) is the area of the given cross-section in terms of x
integrand is a rational function with a factorable denominator


14. Given v(t) find displacement
a^x ln(a)
y' = sec(x)tan(x)
y' = -csc²(x)
? v(t) over interval a to b


15. y = cos?¹(x) - y' =

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16. definite integral
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)
use ratio test - set > 1 and solve absolute value equations - check endpoints
has limits a & b - find antiderivative - F(b) - F(a)
product rule


17. Find radius of convergence
use ratio test - set > 1 and solve absolute value equations - radius = center - endpoint
general term = 1/n^p - converges if p > 1
Slope of tangent line at a point - value of derivative at a point
draw short segments representing slope at each point


18. Instantenous Rate of Change
logistic differential equation - M = carrying capacity
Slope of tangent line at a point - value of derivative at a point
A function and it's derivative are in the integrand
Area of trapezoid


19. trapezoidal rule
? v(t) over interval a to b
use trapezoids to evaluate integrals (estimate area)
use ratio test - set > 1 and solve absolute value equations - radius = center - endpoint
? v (dx/dt)² + (dy/dt)² over interval from a to b


20. Indeterminate forms
critical points and endpoints
0/0 - 8/8 - 8*0 - 8 - 8 - 1^8 - 0° - 8°
increasing
Limit as x approaches a of [f(x)-f(a)]/(x-a)


21. nth term test
if terms grow without bound - series diverges
product rule
e^x
1/(b-a) ? f(x) dx on interval a to b


22. left riemann sum
Slope of secant line between two points - use to estimate instantanous rate of change at a point.
use rectangles with left-endpoints to evaluate integral (estimate area)
relative maximum
y' = -csc(x)cot(x)


23. absolute value of velocity
integrand is a rational function with a factorable denominator
speed
p ? r² dx over interval a to b - where r = distance from curve to axis of revolution
point of inflection


24. dP/dt = kP(M - P)
velocity is negative
v(dx/dt)² + (dy/dt)² not an integral!
increasing
logistic differential equation - M = carrying capacity


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

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

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

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28. y = cot?¹(x) - y' =

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29. [(h1 - h2)/2]*base
y' = -csc²(x)
A function and it's derivative are in the integrand
Area of trapezoid
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)


30. Given v(t) find total distance travelled
relative minimum
use ratio test - set > 1 and solve absolute value equations - check endpoints
? abs[v(t)] over interval a to b
Limit as x approaches a of [f(x)-f(a)]/(x-a)


31. When is a function not differentiable
y' = -1/v(1 - x²)
polynomial with finite number of terms - largest exponent is 6 - find all derivatives up to the 6th derivative
e^x
corner - cusp - vertical tangent - discontinuity


32. To draw a slope field - plug (x -y) coordinates into differential equation...
(uv'-vu')/v²
draw short segments representing slope at each point
point of inflection
use rectangles with right-endpoints to evaluate integrals (estimate area)


33. Find interval of convergence
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
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.
a^x ln(a)


34. Average Rate of Change
Slope of secant line between two points - use to estimate instantanous rate of change at a point.
velocity is negative
product rule
point of inflection


35. When f '(x) is positive - f(x) is...
(uv'-vu')/v²
lim as n approaches 8 of ratio of (n+1) term/nth term > 1 - series converges
increasing
use ratio test - set > 1 and solve absolute value equations - check endpoints


36. y = a^x - y' = y' =
f(x) has a relative maximum
a^x ln(a)
substitution - parts - partial fractions
if lim as n approaches 8 of ratio of comparison series/general term is positive and finite - then series behaves like comparison series


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

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38. y = ln(x)/x² - state rule used to find derivative
? abs[v(t)] over interval a to b
quotient rule
has limits a & b - find antiderivative - F(b) - F(a)
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)


39. Given velocity vectors dx/dt and dy/dt - find speed
f(x) has a relative minimum
concave up
Area of trapezoid
v(dx/dt)² + (dy/dt)² not an integral!


40. y = sin?¹(x) - y' =

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41. y = cos²(3x)
chain rule
concave up
? f(x) - g(x) over interval a to b - where f(x) is top function and g(x) is bottom function
integrand is a rational function with a factorable denominator


42. To find absolute maximum on closed interval [a - b] - you must consider...
critical points and endpoints
lim as n approaches 8 of ratio of (n+1) term/nth term > 1 - series converges
f(x)
Alternating series converges and general term converges with another test


43. Length of parametric curve
v(dx/dt)² + (dy/dt)² not an integral!
? v (dx/dt)² + (dy/dt)² over interval from a to b
y' = sec²(x)
y' = -1/v(1 - x²)


44. When f '(x) changes from negative to positive - f(x) has a...
f '(g(x)) g'(x)
y' = -csc(x)cot(x)
relative minimum
uv' + vu'


45. L'Hopitals rule
use to find indeterminate limits - find derivative of numerator and denominator separately then evaluate limit
Alternating series converges and general term converges with another test
y' = cos(x)
f(x) has a relative minimum


46. area under a curve
use tangent line to approximate values of the function
? f(x) dx integrate over interval a to b
f(x) has a relative minimum
y' = 1/v(1 - x²)


47. Volume of solid with base in the plane and given cross-section
if lim as n approaches 8 of ratio of comparison series/general term is positive and finite - then series behaves like comparison series
find first derivative - dy/dx = dy/dt / dx/dt - then find derivative of first derivative - then divide by dx/dt
? A(x) dx over interval a to b - where A(x) is the area of the given cross-section in terms of x
use rectangles with left-endpoints to evaluate integral (estimate area)


48. use integration by parts when...
no limits - find antiderivative + C - use inital value to find C
relative maximum
two different types of functions are multiplied
? f(x) - g(x) over interval a to b - where f(x) is top function and g(x) is bottom function


49. Area inside one polar curve and outside another polar curve
y' = -csc(x)cot(x)
concave down
y' = 1/(x lna)
1/2 ? R² - r² over interval from a to b - find a & b by setting equations equal - solve for theta.


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

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