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(x) - y' =

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2. 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.
separate variables - integrate + C - use initial condition to find C - solve for y
Slope of secant line between two points - use to estimate instantanous rate of change at a point.
no limits - find antiderivative + C - use inital value to find C


3. Area between two curves
? 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
critical points and endpoints
relative minimum


4. indefinite integral
no limits - find antiderivative + C - use inital value to find C
positive
general term = a1r^n - converges if -1 < r < 1
chain rule


5. When f '(x) is decreasing - f(x) is...
y' = -1/v(1 - x²)
y' = sec²(x)
f(x)
concave down


6. Geometric series test
f(x) has a relative maximum
? f(x) - g(x) over interval a to b - where f(x) is top function and g(x) is bottom function
y' = -csc²(x)
general term = a1r^n - converges if -1 < r < 1


7. y = ln(x)/x² - state rule used to find derivative
quotient rule
y' = cos(x)
1/(b-a) ? f(x) dx on interval a to b
? f(x) dx integrate over interval a to b


8. Given v(t) find displacement
derivative
? v (dx/dt)² + (dy/dt)² over interval from a to b
? v(t) over interval a to b
Area of trapezoid


9. rate
use rectangles with left-endpoints to evaluate integral (estimate area)
concave down
derivative
use rectangles with right-endpoints to evaluate integrals (estimate area)


10. Intermediate Value Theorem
e^x
f(x) has a relative maximum
product rule
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. y = csc(x) - y' =

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12. Particle is moving to the right/up
critical points and endpoints
y' = -1/v(1 - x²)
velocity is positive
y' = 1/x


13. Area inside polar curve
1/2 ? r² over interval from a to b - find a & b by setting r = 0 - solve for theta
y' = -1/v(1 - x²)
increasing
point of inflection


14. Average Rate of Change
? A(x) dx over interval a to b - where A(x) is the area of the given cross-section in terms of x
if lim as n approaches 8 of ratio of comparison series/general term is positive and finite - then series behaves like comparison series
if terms grow without bound - series diverges
Slope of secant line between two points - use to estimate instantanous rate of change at a point.


15. Find interval of convergence
y' = 1/v(1 - x²)
use ratio test - set > 1 and solve absolute value equations - check endpoints
? v (dx/dt)² + (dy/dt)² over interval from a to b
logistic growth equation


16. L'Hopitals rule
use to find indeterminate limits - find derivative of numerator and denominator separately then evaluate limit
f(x) has a relative maximum
1/2 ? r² over interval from a to b - find a & b by setting r = 0 - solve for theta
v(dx/dt)² + (dy/dt)² not an integral!


17. 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
corner - cusp - vertical tangent - discontinuity
? f(x) dx on interval a to b = F(b) - F(a)
a^x ln(a)


18. 6th degree Taylor Polynomial
increasing
draw short segments representing slope at each point
polynomial with finite number of terms - largest exponent is 6 - find all derivatives up to the 6th derivative
0/0 - 8/8 - 8*0 - 8 - 8 - 1^8 - 0° - 8°


19. When f '(x) is positive - f(x) is...
logistic differential equation - M = carrying capacity
find first derivative - dy/dx = dy/dt / dx/dt - then find derivative of first derivative - then divide by dx/dt
increasing
uv - ? v du


20. P = M / (1 + Ae^(-Mkt))
logistic growth equation
has limits a & b - find antiderivative - F(b) - F(a)
f '(g(x)) g'(x)
f(x)


21. Linearization
undefined
y' = -1/(1 + x²)
Slope of secant line between two points - use to estimate instantanous rate of change at a point.
use tangent line to approximate values of the function


22. absolute value of velocity
speed
f(x)
y' = -1/(1 + x²)
1/(b-a) ? f(x) dx on interval a to b


23. y = x cos(x) - state rule used to find derivative
use to find indeterminate limits - find derivative of numerator and denominator separately then evaluate limit
two different types of functions are multiplied
use tangent line to approximate values of the function
product rule


24. [(h1 - h2)/2]*base
uv - ? v du
quotient rule
Area of trapezoid
use rectangles with right-endpoints to evaluate integrals (estimate area)


25. Alternating series tes
substitution - parts - partial fractions
concave up
lim as n approaches zero of general term = 0 and terms decrease - series converges
f '(g(x)) g'(x)


26. Given velocity vectors dx/dt and dy/dt - find total distance travelled
use trapezoids to evaluate integrals (estimate area)
? v(t) over interval a to b
relative maximum
? v (dx/dt)² + (dy/dt)² over interval from a to b


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

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28. slope of horizontal line
? v (dx/dt)² + (dy/dt)² over interval from a to b
use ratio test - set > 1 and solve absolute value equations - radius = center - endpoint
zero
undefined


29. When f '(x) is negative - f(x) is...
f '(g(x)) g'(x)
p ? r² dx over interval a to b - where r = distance from curve to axis of revolution
decreasing
has limits a & b - find antiderivative - F(b) - F(a)


30. Limit comparison test
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
if lim as n approaches 8 of ratio of comparison series/general term is positive and finite - then series behaves like comparison series
general term = 1/n^p - converges if p > 1


31. nth term test
if terms grow without bound - series diverges
y' = -1/v(1 - x²)
f '(g(x)) g'(x)
Slope of tangent line at a point - value of derivative at a point


32. When is a function not differentiable
find first derivative - dy/dx = dy/dt / dx/dt - then find derivative of first derivative - then divide by dx/dt
corner - cusp - vertical tangent - discontinuity
y' = 1/x
velocity is negative


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

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34. Converges absolutely
(uv'-vu')/v²
polynomial with infinite number of terms - includes general term
decreasing
Alternating series converges and general term converges with another test


35. If f '(x) = 0 and f'(x) > 0 -...
f(x)
polynomial with finite number of terms - largest exponent is 6 - find all derivatives up to the 6th derivative
(uv'-vu')/v²
f(x) has a relative minimum


36. Length of curve
? v(1 + (dy/dx)²) dx over interval a to b
A function and it's derivative are in the integrand
draw short segments representing slope at each point
logistic growth equation


37. use substitution to integrate when

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38. area under a curve
? v (dx/dt)² + (dy/dt)² over interval from a to b
if lim as n approaches 8 of ratio of comparison series/general term is positive and finite - then series behaves like comparison 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) dx integrate over interval a to b


39. mean value theorem

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

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41. Find radius of convergence
a^x ln(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.
use ratio test - set > 1 and solve absolute value equations - radius = center - endpoint
if integral converges - series converges


42. Chain Rule

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43. Instantenous Rate of Change
chain rule
use rectangles with right-endpoints to evaluate integrals (estimate area)
logistic differential equation - M = carrying capacity
Slope of tangent line at a point - value of derivative at a point


44. To find particular solution to differential equation - dy/dx = x/y...
p ? r² dx over interval a to b - where r = distance from curve to axis of revolution
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)
1/2 ? R² - r² over interval from a to b - find a & b by setting equations equal - solve for theta.
separate variables - integrate + C - use initial condition to find C - solve for y


45. use integration by parts when...
lim as n approaches 8 of ratio of (n+1) term/nth term > 1 - series converges
? f(x) - g(x) over interval a to b - where f(x) is top function and g(x) is bottom function
two different types of functions are multiplied
product rule


46. Length of parametric curve
? f(x) dx integrate over interval a to b
y' = 1/v(1 - x²)
? v (dx/dt)² + (dy/dt)² over interval from a to b
A function and it's derivative are in the integrand


47. y = a^x - y' = y' =
? v(1 + (dy/dx)²) dx over interval a to b
a^x ln(a)
use rectangles with right-endpoints to evaluate integrals (estimate area)
? v (dx/dt)² + (dy/dt)² over interval from a to b


48. y = tan?¹(x) - y' =

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49. p-series test
general term = 1/n^p - converges if p > 1
Alternating series converges and general term converges with another test
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.


50. Integral test
speed
? 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
f(x) has a relative maximum