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. 6th degree Taylor Polynomial
1/2 ? R² - r² over interval from a to b - find a & b by setting equations equal - solve for theta.
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polynomial with finite number of terms - largest exponent is 6 - find all derivatives up to the 6th derivative
a^x ln(a)


2. Area inside polar curve
? v (dx/dt)² + (dy/dt)² over interval from a to b
uv' + vu'
1/2 ? r² over interval from a to b - find a & b by setting r = 0 - solve for theta
general term = a1r^n - converges if -1 < r < 1


3. Find radius of convergence
? v (dx/dt)² + (dy/dt)² over interval from a to b
two different types of functions are multiplied
0/0 - 8/8 - 8*0 - 8 - 8 - 1^8 - 0° - 8°
use ratio test - set > 1 and solve absolute value equations - radius = center - endpoint


4. If g(x) = ? f(t) dt on interval 2 to x - then g'(x) =...
y' = -csc(x)cot(x)
v(dx/dt)² + (dy/dt)² not an integral!
separate variables - integrate + C - use initial condition to find C - solve for y
f(x)


5. Chain Rule

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6. Volume of solid of revolution - washer
? f(x) - g(x) over interval a to b - where f(x) is top function and g(x) is bottom function
y' = -sin(x)
lim as n approaches zero of general term = 0 and terms decrease - 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


7. slope of vertical line
Limit as x approaches a of [f(x)-f(a)]/(x-a)
? f(x) dx integrate over interval a to b
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1/(b-a) ? f(x) dx on interval a to b


8. Use partial fractions to integrate when...
polynomial with infinite number of terms - includes general term
derivative
? v (dx/dt)² + (dy/dt)² over interval from a to b
integrand is a rational function with a factorable denominator


9. When f '(x) changes from increasing to decreasing or decreasing to increasing - f(x) has a...
Area of trapezoid
decreasing
point of inflection
A function and it's derivative are in the integrand


10. Geometric series test
speed
Alternating series converges and general term converges with another test
use ratio test - set > 1 and solve absolute value equations - check endpoints
general term = a1r^n - converges if -1 < r < 1


11. Taylor series
polynomial with infinite number of terms - includes general term
y' = 1/(1 + x²)
1/2 ? R² - r² over interval from a to b - find a & b by setting equations equal - solve for theta.
y' = -1/v(1 - x²)


12. Linearization
has limits a & b - find antiderivative - F(b) - F(a)
y' = 1/x
logistic growth equation
use tangent line to approximate values of the function


13. rate
substitution - parts - partial fractions
derivative
use tangent line to approximate values of the function
general term = a1r^n - converges if -1 < r < 1


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

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15. y = tan?¹(x) - y' =

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16. Average Rate of Change
Slope of secant line between two points - use to estimate instantanous rate of change at a point.
? f(x) - g(x) over interval a to b - where f(x) is top function and g(x) is bottom function
y' = -1/(1 + x²)
polynomial with finite number of terms - largest exponent is 6 - find all derivatives up to the 6th derivative


17. Instantenous Rate of Change
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concave up
Slope of tangent line at a point - value of derivative at a point
general term = a1r^n - converges if -1 < r < 1


18. average value of f(x)
(uv'-vu')/v²
use rectangles with left-endpoints to evaluate integral (estimate area)
1/(b-a) ? f(x) dx on interval a to b
polynomial with infinite number of terms - includes general term


19. When f '(x) is positive - f(x) is...
concave up
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increasing
f '(g(x)) g'(x)


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

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21. Volume of solid of revolution - no washer
p ? r² dx over interval a to b - where r = distance from curve to axis of revolution
velocity is positive
general term = 1/n^p - converges if p > 1
no limits - find antiderivative + C - use inital value to find C


22. p-series test
use trapezoids to evaluate integrals (estimate area)
use to find indeterminate limits - find derivative of numerator and denominator separately then evaluate limit
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.
general term = 1/n^p - converges if p > 1


23. mean value theorem

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24. left riemann sum
use rectangles with left-endpoints to evaluate integral (estimate area)
p ? r² dx over interval a to b - where r = distance from curve to axis of revolution
? f(x) dx on interval a to b = F(b) - F(a)
? v(1 + (dy/dx)²) dx over interval a to b


25. Length of parametric curve
Limit as x approaches a of [f(x)-f(a)]/(x-a)
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
use rectangles with right-endpoints to evaluate integrals (estimate area)


26. 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.
v(dx/dt)² + (dy/dt)² not an integral!
if lim as n approaches 8 of ratio of comparison series/general term is positive and finite - then series behaves like comparison series
y' = -1/v(1 - x²)


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

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28. Given velocity vectors dx/dt and dy/dt - find total distance travelled
1/2 ? r² over interval from a to b - find a & b by setting r = 0 - solve for theta
has limits a & b - find antiderivative - F(b) - F(a)
positive
? v (dx/dt)² + (dy/dt)² over interval from a to b


29. Integral test
if integral converges - series converges
f '(g(x)) g'(x)
? v (dx/dt)² + (dy/dt)² over interval from a to b
? A(x) dx over interval a to b - where A(x) is the area of the given cross-section in terms of x


30. slope of horizontal line
critical points and endpoints
general term = 1/n^p - converges if p > 1
concave up
zero


31. y = e^x - y' = y' =
e^x
general term = 1/n^p - converges if p > 1
velocity is positive
? v (dx/dt)² + (dy/dt)² over interval from a to b


32. methods of integration
use trapezoids to evaluate integrals (estimate area)
substitution - parts - partial fractions
? f(x) - g(x) over interval a to b - where f(x) is top function and g(x) is bottom function
derivative


33. When f '(x) changes fro positive to negative - f(x) has a...
? v (dx/dt)² + (dy/dt)² over interval from a to b
negative
point of inflection
relative maximum


34. Limit comparison test
if lim as n approaches 8 of ratio of comparison series/general term is positive and finite - then series behaves like comparison series
zero
use to find indeterminate limits - find derivative of numerator and denominator separately then evaluate limit
y' = 1/v(1 - x²)


35. Formal definition of derivative
uv - ? v du
Limit as h approaches 0 of [f(a+h)-f(a)]/h
positive
? abs[v(t)] over interval a to b


36. nth term test
Slope of secant line between two points - use to estimate instantanous rate of change at a point.
if terms grow without bound - series diverges
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? abs[v(t)] over interval a to b


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

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38. If f '(x) = 0 and f'(x) > 0 -...
f(x) has a relative minimum
A function and it's derivative are in the integrand
velocity is negative
concave down


39. y = sin?¹(x) - y' =

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40. area under a curve
? f(x) dx integrate over interval a to b
if terms grow without bound - series diverges
product rule
if integral converges - series converges


41. If f '(x) = 0 and f'(x) < 0 -...
y' = 1/x
use to find indeterminate limits - find derivative of numerator and denominator separately then evaluate limit
f(x) has a relative maximum
product rule


42. indefinite integral
1/2 ? r² over interval from a to b - find a & b by setting r = 0 - solve for theta
y' = -1/v(1 - x²)
use tangent line to approximate values of the function
no limits - find antiderivative + C - use inital value to find C


43. use integration by parts when...
two different types of functions are multiplied
relative maximum
general term = 1/n^p - converges if p > 1
polynomial with infinite number of terms - includes general term


44. Alternate definition of derivative
speed
Limit as x approaches a of [f(x)-f(a)]/(x-a)
Area of trapezoid
? A(x) dx over interval a to b - where A(x) is the area of the given cross-section in terms of x


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

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46. use substitution to integrate when

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47. Given v(t) find displacement
e^x
? v(t) over interval a to b
y' = sec(x)tan(x)
1/(b-a) ? f(x) dx on interval a to b


48. When f '(x) is negative - f(x) is...
1/(b-a) ? f(x) dx on interval a to b
decreasing
? v(t) over interval a to b
y' = -1/(1 + x²)


49. Converges conditionally
Area of trapezoid
lim as n approaches zero of general term = 0 and terms decrease - series converges
relative maximum
Alternating series converges and general term diverges with another test


50. Intermediate Value Theorem
? f(x) dx integrate over interval a to b
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relative maximum
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.