AP Calculus Bc

Instructions:

• Answer 50 questions in 15 minutes.
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• 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. Formal definition of derivative
positive
Limit as h approaches 0 of [f(a+h)-f(a)]/h
velocity is positive
lim as n approaches zero of general term = 0 and terms decrease - series converges


2. trapezoidal rule
use trapezoids to evaluate integrals (estimate area)
y' = -csc²(x)
? v(t) over interval a to b
? f(x) dx on interval a to b = F(b) - F(a)


3. Find radius of convergence
integrand is a rational function with a factorable denominator
use ratio test - set > 1 and solve absolute value equations - radius = center - endpoint
polynomial with infinite number of terms - includes general term
if terms grow without bound - series diverges


4. To find absolute maximum on closed interval [a - b] - you must consider...
corner - cusp - vertical tangent - discontinuity
? f(x) dx integrate over interval a to b
critical points and endpoints
speed


5. slope of horizontal line
? f(x) - g(x) over interval a to b - where f(x) is top function and g(x) is bottom function
A function and it's derivative are in the integrand
? v (dx/dt)² + (dy/dt)² over interval from a to b
zero


6. Volume of solid with base in the plane and given cross-section
f '(g(x)) g'(x)
point of inflection
y' = 1/(1 + x²)
? A(x) dx over interval a to b - where A(x) is the area of the given cross-section in terms of x


7. p-series test
general term = 1/n^p - converges if p > 1
logistic differential equation - M = carrying capacity
? f(x) dx integrate over interval a to b
product rule


8. Given v(t) find total distance travelled
quotient rule
e^x
? v (dx/dt)² + (dy/dt)² over interval from a to b
? abs[v(t)] over interval a to b


9. [(h1 - h2)/2]*base
if integral converges - series converges
increasing
use rectangles with left-endpoints to evaluate integral (estimate area)
Area of trapezoid


10. When f '(x) changes from increasing to decreasing or decreasing to increasing - f(x) has a...
? f(x) dx integrate over interval a to b
two different types of functions are multiplied
point of inflection
zero


11. Eatio test
Alternating series converges and general term converges with another test
lim as n approaches 8 of ratio of (n+1) term/nth term > 1 - series converges
y' = -1/v(1 - x²)
speed


12. left riemann sum
y' = -csc²(x)
use rectangles with left-endpoints to evaluate integral (estimate area)
has limits a & b - find antiderivative - F(b) - F(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.


13. Particle is moving to the right/up
speed
v(dx/dt)² + (dy/dt)² not an integral!
polynomial with finite number of terms - largest exponent is 6 - find all derivatives up to the 6th derivative
velocity is positive


14. use integration by parts when...
two different types of functions are multiplied
e^x
concave up
? v(1 + (dy/dx)²) dx over interval a to b


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

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16. mean value theorem

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17. If g(x) = ? f(t) dt on interval 2 to x - then g'(x) =...
f(x)
point of inflection
use tangent line to approximate values of the function
separate variables - integrate + C - use initial condition to find C - solve for y


18. Chain Rule

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19. Given velocity vectors dx/dt and dy/dt - find speed
velocity is negative
Limit as x approaches a of [f(x)-f(a)]/(x-a)
general term = a1r^n - converges if -1 < r < 1
v(dx/dt)² + (dy/dt)² not an integral!


20. Find interval of convergence
no limits - find antiderivative + C - use inital value to find C
use ratio test - set > 1 and solve absolute value equations - check endpoints
if lim as n approaches 8 of ratio of comparison series/general term is positive and finite - then series behaves like comparison series
Slope of tangent line at a point - value of derivative at a point


21. 6th degree Taylor Polynomial
polynomial with finite number of terms - largest exponent is 6 - find all derivatives up to the 6th derivative
y' = -1/v(1 - x²)
p ? r² dx over interval a to b - where r = distance from curve to axis of revolution
concave down


22. Given v(t) find displacement
velocity is negative
? v(t) over interval a to b
logistic growth equation
separate variables - integrate + C - use initial condition to find C - solve for y


23. Alternate definition of derivative
critical points and endpoints
? v(1 + (dy/dx)²) dx over interval a to b
uv - ? v du
Limit as x approaches a of [f(x)-f(a)]/(x-a)


24. Integral test
? f(x) - g(x) over interval a to b - where f(x) is top function and g(x) is bottom function
if integral converges - series converges
? abs[v(t)] over interval a to b
quotient rule


25. definite integral
positive
has limits a & b - find antiderivative - F(b) - F(a)
y' = -sin(x)
if lim as n approaches 8 of ratio of comparison series/general term is positive and finite - then series behaves like comparison series


26. Area inside one polar curve and outside another polar curve
lim as n approaches 8 of ratio of (n+1) term/nth term > 1 - series converges
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.
Limit as h approaches 0 of [f(a+h)-f(a)]/h
1/2 ? R² - r² over interval from a to b - find a & b by setting equations equal - solve for theta.


27. indefinite integral
? v(t) over interval a to b
no limits - find antiderivative + C - use inital value to find C
y' = sec(x)tan(x)
product rule


28. When f '(x) changes fro positive to negative - f(x) has a...
velocity is positive
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 maximum
uv' + vu'


29. y = a^x - y' = y' =
a^x ln(a)
? v(t) over interval a to b
if integral converges - series converges
A function and it's derivative are in the integrand


30. y = tan?¹(x) - y' =

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

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

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33. Particle is moving to the left/down
? A(x) dx over interval a to b - where A(x) is the area of the given cross-section in terms of x
Limit as h approaches 0 of [f(a+h)-f(a)]/h
velocity is negative
if terms grow without bound - series diverges


34. When f '(x) changes from negative to positive - f(x) has a...
relative minimum
point of inflection
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


35. Fundamental Theorem of Calculus
? f(x) dx on interval a to b = F(b) - F(a)
? v (dx/dt)² + (dy/dt)² over interval from a to b
Alternating series converges and general term converges with another test
Limit as x approaches a of [f(x)-f(a)]/(x-a)


36. Area between two curves
? 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 right-endpoints to evaluate integrals (estimate area)
Limit as x approaches a of [f(x)-f(a)]/(x-a)
? f(x) - g(x) over interval a to b - where f(x) is top function and g(x) is bottom function


37. Intermediate Value Theorem
general term = 1/n^p - converges if p > 1
y' = sec(x)tan(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.
? A(x) dx over interval a to b - where A(x) is the area of the given cross-section in terms of x


38. If f '(x) = 0 and f'(x) < 0 -...
f(x) has a relative maximum
has limits a & b - find antiderivative - F(b) - F(a)
negative
y' = 1/x


39. L'Hopitals rule
relative maximum
negative
quotient rule
use to find indeterminate limits - find derivative of numerator and denominator separately then evaluate limit


40. y = x cos(x) - state rule used to find derivative
y' = -sin(x)
velocity is positive
product rule
? f(x) dx on interval a to b = F(b) - F(a)


41. Length of parametric curve
f(x) has a relative minimum
? v (dx/dt)² + (dy/dt)² over interval from a to b
f(x) has a relative maximum
Slope of tangent line at a point - value of derivative at a point


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

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43. Limit comparison test
polynomial with infinite number of terms - includes general term
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
relative maximum


44. Geometric series test
logistic growth equation
lim as n approaches 8 of ratio of (n+1) term/nth term > 1 - series converges
f '(g(x)) g'(x)
general term = a1r^n - converges if -1 < r < 1


45. When f '(x) is positive - f(x) is...
concave up
y' = cos(x)
y' = 1/(1 + x²)
increasing


46. Converges absolutely
Alternating series converges and general term converges with another test
two different types of functions are multiplied
velocity is positive
concave up


47. When f '(x) is increasing - f(x) is...
point of inflection
use trapezoids to evaluate integrals (estimate area)
concave up
concave down


48. Taylor series
? A(x) dx over interval a to b - where A(x) is the area of the given cross-section in terms of x
f(x) has a relative minimum
a^x ln(a)
polynomial with infinite number of terms - includes general term


49. When f '(x) is decreasing - f(x) is...
concave down
Limit as x approaches a of [f(x)-f(a)]/(x-a)
derivative
use ratio test - set > 1 and solve absolute value equations - radius = center - endpoint


50. Use partial fractions to integrate when...
y' = -1/(1 + x²)
if terms grow without bound - series diverges
p ? r² dx over interval a to b - where r = distance from curve to axis of revolution
integrand is a rational function with a factorable denominator