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. y = sin?¹(x) - y' =

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2. y = ln(x)/x² - state rule used to find derivative
if lim as n approaches 8 of ratio of comparison series/general term is positive and finite - then series behaves like comparison series
increasing
quotient rule
use to find indeterminate limits - find derivative of numerator and denominator separately then evaluate limit


3. Use partial fractions to integrate when...
polynomial with infinite number of terms - includes general term
positive
integrand is a rational function with a factorable denominator
critical points and endpoints


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

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5. Taylor series
use rectangles with left-endpoints to evaluate integral (estimate area)
0/0 - 8/8 - 8*0 - 8 - 8 - 1^8 - 0° - 8°
y' = -sin(x)
polynomial with infinite number of terms - includes general term


6. average value of f(x)
product rule
substitution - parts - partial fractions
1/(b-a) ? f(x) dx on interval a to b
y' = -sin(x)


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

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8. To draw a slope field - plug (x -y) coordinates into differential equation...
p ? r² dx over interval a to b - where r = distance from curve to axis of revolution
general term = 1/n^p - converges if p > 1
draw short segments representing slope at each point
? f(x) dx on interval a to b = F(b) - F(a)


9. slope of horizontal line
y' = sec(x)tan(x)
? A(x) dx over interval a to b - where A(x) is the area of the given cross-section in terms of x
increasing
zero


10. Intermediate Value Theorem
logistic differential equation - M = carrying capacity
? v (dx/dt)² + (dy/dt)² over interval from a to b
general term = a1r^n - converges if -1 < r < 1
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.


11. Area inside polar curve
logistic growth equation
1/2 ? r² over interval from a to b - find a & b by setting r = 0 - solve for theta
(uv'-vu')/v²
velocity is negative


12. If f '(x) = 0 and f'(x) < 0 -...
logistic growth equation
f(x) has a relative maximum
y' = -csc²(x)
? v(1 + (dy/dx)²) dx over interval a to b


13. Linearization
use tangent line to approximate values of the function
increasing
A function and it's derivative are in the integrand
Alternating series converges and general term diverges with another test


14. Find interval of convergence
lim as n approaches 8 of ratio of (n+1) term/nth term > 1 - series converges
use tangent line to approximate values of the function
use ratio test - set > 1 and solve absolute value equations - check endpoints
chain rule


15. Indeterminate forms
0/0 - 8/8 - 8*0 - 8 - 8 - 1^8 - 0° - 8°
? A(x) dx over interval a to b - where A(x) is the area of the given cross-section in terms of x
f '(g(x)) g'(x)
use trapezoids to evaluate integrals (estimate area)


16. Product Rule

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17. Quotient Rule

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18. Converges conditionally
use rectangles with left-endpoints to evaluate integral (estimate area)
relative minimum
Alternating series converges and general term diverges with another test
Limit as x approaches a of [f(x)-f(a)]/(x-a)


19. nth term test
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/(x lna)
point of inflection
if terms grow without bound - series diverges


20. slope of vertical line
undefined
polynomial with finite number of terms - largest exponent is 6 - find all derivatives up to the 6th derivative
1/2 ? R² - r² over interval from a to b - find a & b by setting equations equal - solve for theta.
substitution - parts - partial fractions


21. Volume of solid with base in the plane and given cross-section
polynomial with finite number of terms - largest exponent is 6 - find all derivatives up to the 6th derivative
? A(x) dx over interval a to b - where A(x) is the area of the given cross-section in terms of x
y' = 1/(1 + x²)
critical points and endpoints


22. use integration by parts when...
two different types of functions are multiplied
increasing
Area of trapezoid
v(dx/dt)² + (dy/dt)² not an integral!


23. Given velocity vectors dx/dt and dy/dt - find speed
lim as n approaches 8 of ratio of (n+1) term/nth term > 1 - series converges
product rule
v(dx/dt)² + (dy/dt)² not an integral!
substitution - parts - partial fractions


24. When is a function not differentiable
Limit as x approaches a of [f(x)-f(a)]/(x-a)
use rectangles with left-endpoints to evaluate integral (estimate area)
use trapezoids to evaluate integrals (estimate area)
corner - cusp - vertical tangent - discontinuity


25. Second derivative of parametrically defined curve
find first derivative - dy/dx = dy/dt / dx/dt - then find derivative of first derivative - then divide by dx/dt
product rule
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
zero


26. Fundamental Theorem of Calculus
f(x) has a relative maximum
draw short segments representing slope at each point
? f(x) dx on interval a to b = F(b) - F(a)
p ? r² dx over interval a to b - where r = distance from curve to axis of revolution


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

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28. y = e^x - y' = y' =
corner - cusp - vertical tangent - discontinuity
e^x
no limits - find antiderivative + C - use inital value to find C
draw short segments representing slope at each point


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

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

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31. Instantenous Rate of Change
f(x) has a relative minimum
Slope of tangent line at a point - value of derivative at a point
use ratio test - set > 1 and solve absolute value equations - check endpoints
relative maximum


32. Given velocity vectors dx/dt and dy/dt - find total distance travelled
no limits - find antiderivative + C - use inital value to find C
A function and it's derivative are in the integrand
f(x) has a relative maximum
? v (dx/dt)² + (dy/dt)² over interval from a to b


33. When f '(x) changes from increasing to decreasing or decreasing to increasing - f(x) has a...
no limits - find antiderivative + C - use inital value to find C
point of inflection
if terms grow without bound - series diverges
y' = sec²(x)


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

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35. L'Hopitals rule
use to find indeterminate limits - find derivative of numerator and denominator separately then evaluate limit
0/0 - 8/8 - 8*0 - 8 - 8 - 1^8 - 0° - 8°
use rectangles with right-endpoints to evaluate integrals (estimate area)
y' = 1/v(1 - x²)


36. Chain Rule

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37. rate
derivative
p ? r² dx over interval a to b - where r = distance from curve to axis of revolution
? v(t) over interval a to b
1/2 ? R² - r² over interval from a to b - find a & b by setting equations equal - solve for theta.


38. Given v(t) find displacement
? v(t) over interval a to b
y' = sec²(x)
relative minimum
Slope of secant line between two points - use to estimate instantanous rate of change at a point.


39. definite integral
has limits a & b - find antiderivative - F(b) - F(a)
if integral converges - series converges
Limit as h approaches 0 of [f(a+h)-f(a)]/h
f '(g(x)) g'(x)


40. Volume of solid of revolution - no washer
p ? r² dx over interval a to b - where r = distance from curve to axis of revolution
general term = a1r^n - converges if -1 < r < 1
a^x ln(a)
y' = -1/v(1 - x²)


41. absolute value of velocity
zero
undefined
quotient rule
speed


42. Volume of solid of revolution - washer
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
? v (dx/dt)² + (dy/dt)² over interval from a to b
substitution - parts - partial fractions
1/2 ? R² - r² over interval from a to b - find a & b by setting equations equal - solve for theta.


43. y = x cos(x) - state rule used to find derivative
use rectangles with right-endpoints to evaluate integrals (estimate area)
y' = 1/x
product rule
Area of trapezoid


44. When f '(x) is increasing - f(x) is...
1/2 ? r² over interval from a to b - find a & b by setting r = 0 - solve for theta
? A(x) dx over interval a to b - where A(x) is the area of the given cross-section in terms of x
concave up
velocity is positive


45. Geometric series test
general term = a1r^n - converges if -1 < r < 1
positive
negative
? f(x) dx integrate over interval a to b


46. area above x-axis is...
draw short segments representing slope at each point
positive
no limits - find antiderivative + C - use inital value to find C
? v (dx/dt)² + (dy/dt)² over interval from a to b


47. When f '(x) changes from negative to positive - f(x) has a...
Alternating series converges and general term diverges with another test
y' = sec(x)tan(x)
use ratio test - set > 1 and solve absolute value equations - check endpoints
relative minimum


48. area below x-axis is...
y' = -sin(x)
undefined
negative
separate variables - integrate + C - use initial condition to find C - solve for y


49. [(h1 - h2)/2]*base
y' = -csc(x)cot(x)
has limits a & b - find antiderivative - F(b) - F(a)
y' = -1/v(1 - x²)
Area of trapezoid


50. Integral test
use rectangles with left-endpoints to evaluate integral (estimate area)
if integral converges - series converges
use to find indeterminate limits - find derivative of numerator and denominator separately then evaluate limit
e^x