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. area above x-axis is...
positive
undefined
A function and it's derivative are in the integrand
polynomial with finite number of terms - largest exponent is 6 - find all derivatives up to the 6th derivative


2. Average Rate of Change
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
Slope of secant line between two points - use to estimate instantanous rate of change at a point.
general term = a1r^n - converges if -1 < r < 1


3. Eatio test
lim as n approaches 8 of ratio of (n+1) term/nth term > 1 - series converges
polynomial with finite number of terms - largest exponent is 6 - find all derivatives up to the 6th derivative
? v(t) over interval a to b
Limit as h approaches 0 of [f(a+h)-f(a)]/h


4. L'Hopitals rule
use to find indeterminate limits - find derivative of numerator and denominator separately then evaluate limit
y' = cos(x)
a^x ln(a)
y' = 1/(1 + x²)


5. rate
lim as n approaches 8 of ratio of (n+1) term/nth term > 1 - series converges
derivative
use trapezoids to evaluate integrals (estimate area)
product rule


6. Formal definition of derivative
a^x ln(a)
logistic differential equation - M = carrying capacity
relative maximum
Limit as h approaches 0 of [f(a+h)-f(a)]/h


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

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8. Second derivative of parametrically defined curve
Slope of tangent line at a point - value of derivative at a point
find first derivative - dy/dx = dy/dt / dx/dt - then find derivative of first derivative - then divide by dx/dt
use rectangles with right-endpoints to evaluate integrals (estimate area)
? A(x) dx over interval a to b - where A(x) is the area of the given cross-section in terms of x


9. area below x-axis is...
? abs[v(t)] over interval a to b
negative
integrand is a rational function with a factorable denominator
speed


10. y = tan?¹(x) - y' =

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11. Integral test
if integral converges - series converges
draw short segments representing slope at each point
if terms grow without bound - series diverges
Area of trapezoid


12. use substitution to integrate when

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13. Volume of solid of revolution - no washer
p ? r² dx over interval a to b - where r = distance from curve to axis of revolution
y' = sec(x)tan(x)
use rectangles with right-endpoints to evaluate integrals (estimate area)
y' = cos(x)


14. right riemann sum
substitution - parts - partial fractions
? v(1 + (dy/dx)²) dx over interval a to b
y' = -1/v(1 - x²)
use rectangles with right-endpoints to evaluate integrals (estimate area)


15. Particle is moving to the left/down
velocity is negative
Area of trapezoid
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.
uv' + vu'


16. Volume of solid of revolution - washer
velocity is positive
use rectangles with right-endpoints to evaluate integrals (estimate area)
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
speed


17. Volume of solid with base in the plane and given cross-section
point of inflection
? f(x) dx integrate over interval 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
speed


18. y = ln(x)/x² - state rule used to find derivative
f(x) has a relative maximum
velocity is negative
if terms grow without bound - series diverges
quotient rule


19. Instantenous Rate of Change
? v (dx/dt)² + (dy/dt)² over interval from a to b
general term = 1/n^p - converges if p > 1
a^x ln(a)
Slope of tangent line at a point - value of derivative at a point


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

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21. use integration by parts when...
? v(t) over interval a to b
? v (dx/dt)² + (dy/dt)² over interval from a to b
logistic differential equation - M = carrying capacity
two different types of functions are multiplied


22. To find particular solution to differential equation - dy/dx = x/y...
relative minimum
if integral converges - series converges
f(x) has a relative minimum
separate variables - integrate + C - use initial condition to find C - solve for y


23. Linearization
use tangent line to approximate values of the function
velocity is positive
point of inflection
general term = a1r^n - converges if -1 < r < 1


24. Geometric series test
? v (dx/dt)² + (dy/dt)² over interval from a to b
corner - cusp - vertical tangent - discontinuity
v(dx/dt)² + (dy/dt)² not an integral!
general term = a1r^n - converges if -1 < r < 1


25. Area inside one polar curve and outside another polar curve
polynomial with infinite number of terms - includes general term
y' = -csc²(x)
1/2 ? R² - r² over interval from a to b - find a & b by setting equations equal - solve for theta.
uv' + vu'


26. Area inside polar curve
1/2 ? r² over interval from a to b - find a & b by setting r = 0 - solve for theta
f(x) has a relative minimum
? f(x) dx on interval a to b = F(b) - F(a)
if lim as n approaches 8 of ratio of comparison series/general term is positive and finite - then series behaves like comparison series


27. y = cot?¹(x) - y' =

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28. P = M / (1 + Ae^(-Mkt))
logistic growth equation
y' = cos(x)
use to find indeterminate limits - find derivative of numerator and denominator separately then evaluate limit
Limit as x approaches a of [f(x)-f(a)]/(x-a)


29. When f '(x) is increasing - f(x) is...
concave up
point of inflection
? f(x) dx integrate over interval a to b
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


30. slope of vertical line
undefined
has limits a & b - find antiderivative - F(b) - F(a)
y' = 1/x
general term = 1/n^p - converges if p > 1


31. Length of parametric curve
no limits - find antiderivative + C - use inital value to find C
if terms grow without bound - series diverges
? v (dx/dt)² + (dy/dt)² over interval from a to b
lim as n approaches 8 of ratio of (n+1) term/nth term > 1 - series converges


32. absolute value of velocity
has limits a & b - find antiderivative - F(b) - F(a)
speed
uv - ? v du
use rectangles with left-endpoints to evaluate integral (estimate area)


33. y = e^x - y' = y' =
Slope of secant line between two points - use to estimate instantanous rate of change at a point.
Area of trapezoid
e^x
use ratio test - set > 1 and solve absolute value equations - radius = center - endpoint


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

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

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

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37. p-series test
use ratio test - set > 1 and solve absolute value equations - radius = center - endpoint
velocity is positive
general term = 1/n^p - converges if p > 1
f '(g(x)) g'(x)


38. Chain Rule

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

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40. 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
relative maximum
use ratio test - set > 1 and solve absolute value equations - check endpoints
1/2 ? r² over interval from a to b - find a & b by setting r = 0 - solve for theta


41. When f '(x) is negative - f(x) is...
integrand is a rational function with a factorable denominator
decreasing
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)
0/0 - 8/8 - 8*0 - 8 - 8 - 1^8 - 0° - 8°


42. When is a function not differentiable
corner - cusp - vertical tangent - discontinuity
critical points and endpoints
concave up
y' = 1/(x lna)


43. y = x cos(x) - state rule used to find derivative
product rule
zero
y' = -csc(x)cot(x)
Area of trapezoid


44. Use partial fractions to integrate when...
A function and it's derivative are in the integrand
y' = 1/(x lna)
integrand is a rational function with a factorable denominator
uv - ? v du


45. Converges conditionally
y' = -csc²(x)
substitution - parts - partial fractions
Alternating series converges and general term diverges with another test
0/0 - 8/8 - 8*0 - 8 - 8 - 1^8 - 0° - 8°


46. Length of curve
general term = 1/n^p - converges if p > 1
Alternating series converges and general term diverges with another test
? v(1 + (dy/dx)²) dx over interval a to b
concave down


47. Quotient Rule

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48. nth term test
0/0 - 8/8 - 8*0 - 8 - 8 - 1^8 - 0° - 8°
if terms grow without bound - series diverges
Slope of tangent line at a point - value of derivative at a point
quotient rule


49. 6th degree Taylor Polynomial
polynomial with finite number of terms - largest exponent is 6 - find all derivatives up to the 6th derivative
e^x
general term = a1r^n - converges if -1 < r < 1
? v(t) over interval a to b


50. When f '(x) is decreasing - f(x) is...
logistic differential equation - M = carrying capacity
concave down
use tangent line to approximate values of the function
? v (dx/dt)² + (dy/dt)² over interval from a to b