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Test your basic knowledge |
AP Calculus Bc
Start Test
Study First
Subjects
:
math
,
ap
,
calculus
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. trapezoidal rule
y' = sec²(x)
0/0 - 8/8 - 8*0 - 8 - 8 - 1^8 - 0° - 8°
? v(t) over interval a to b
use trapezoids to evaluate integrals (estimate area)
2. Length of curve
? v(1 + (dy/dx)²) dx over interval a to b
no limits - find antiderivative + C - use inital value to find C
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.
if integral converges - series converges
3. When is a function not differentiable
1/2 ? r² over interval from a to b - find a & b by setting r = 0 - solve for theta
Alternating series converges and general term diverges with another test
corner - cusp - vertical tangent - discontinuity
use tangent line to approximate values of the function
4. Taylor series
Limit as h approaches 0 of [f(a+h)-f(a)]/h
chain rule
a^x ln(a)
polynomial with infinite number of terms - includes general term
5. Particle is moving to the right/up
undefined
y' = -1/v(1 - x²)
velocity is positive
? v (dx/dt)² + (dy/dt)² over interval from a to b
6. Area inside polar curve
1/2 ? r² over interval from a to b - find a & b by setting r = 0 - solve for theta
substitution - parts - partial fractions
use rectangles with left-endpoints to evaluate integral (estimate area)
(uv'-vu')/v²
7. Length of parametric curve
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
y' = sec²(x)
uv - ? v du
8. Average Rate of Change
Slope of secant line between two points - use to estimate instantanous rate of change at a point.
chain rule
relative minimum
polynomial with finite number of terms - largest exponent is 6 - find all derivatives up to the 6th derivative
9. L'Hopitals rule
y' = 1/x
speed
use to find indeterminate limits - find derivative of numerator and denominator separately then evaluate limit
if integral converges - series converges
10. If f '(x) = 0 and f'(x) > 0 -...
quotient rule
f(x) has a relative minimum
y' = 1/(1 + x²)
? v (dx/dt)² + (dy/dt)² over interval from a to b
11. Given v(t) find displacement
? v(t) over interval a to b
draw short segments representing slope at each point
Limit as x approaches a of [f(x)-f(a)]/(x-a)
y' = -csc(x)cot(x)
12. Given velocity vectors dx/dt and dy/dt - find speed
v(dx/dt)² + (dy/dt)² not an integral!
A function and it's derivative are in the integrand
e^x
has limits a & b - find antiderivative - F(b) - F(a)
13. Product Rule
14. mean value theorem
15. indefinite integral
no limits - find antiderivative + C - use inital value to find C
use to find indeterminate limits - find derivative of numerator and denominator separately then evaluate limit
1/2 ? R² - r² over interval from a to b - find a & b by setting equations equal - solve for theta.
y' = 1/(x lna)
16. Linearization
use tangent line to approximate values of the function
Alternating series converges and general term diverges with another test
f(x)
y' = 1/v(1 - x²)
17. right riemann sum
a^x ln(a)
use rectangles with right-endpoints to evaluate integrals (estimate area)
point of inflection
f(x) has a relative minimum
18. y = cos²(3x)
logistic growth equation
y' = -csc(x)cot(x)
velocity is positive
chain rule
19. P = M / (1 + Ae^(-Mkt))
0/0 - 8/8 - 8*0 - 8 - 8 - 1^8 - 0° - 8°
p ? r² dx over interval a to b - where r = distance from curve to axis of revolution
logistic growth equation
y' = 1/(x lna)
20. [(h1 - h2)/2]*base
0/0 - 8/8 - 8*0 - 8 - 8 - 1^8 - 0° - 8°
y' = 1/v(1 - x²)
Area of trapezoid
if integral converges - series converges
21. When f '(x) is negative - f(x) is...
relative minimum
find first derivative - dy/dx = dy/dt / dx/dt - then find derivative of first derivative - then divide by dx/dt
decreasing
product rule
22. Use partial fractions to integrate when...
separate variables - integrate + C - use initial condition to find C - solve for y
f(x)
? v(1 + (dy/dx)²) dx over interval a to b
integrand is a rational function with a factorable denominator
23. Chain Rule
24. When f '(x) is positive - f(x) is...
? v(t) over interval a to b
1/(b-a) ? f(x) dx on interval a to b
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.
increasing
25. Intermediate Value Theorem
find first derivative - dy/dx = dy/dt / dx/dt - then find derivative of first derivative - then divide by dx/dt
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.
logistic differential equation - M = carrying capacity
two different types of functions are multiplied
26. 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
y' = -sin(x)
Limit as x approaches a of [f(x)-f(a)]/(x-a)
use rectangles with left-endpoints to evaluate integral (estimate area)
27. y = sin?¹(x) - y' =
28. y = cos(x) - y' =
29. To find particular solution to differential equation - dy/dx = x/y...
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)
? abs[v(t)] over interval a to b
separate variables - integrate + C - use initial condition to find C - solve for y
y' = -csc²(x)
30. use integration by parts when...
two different types of functions are multiplied
? v(1 + (dy/dx)²) dx over interval a to b
? v(t) over interval a to b
y' = 1/(1 + x²)
31. Particle is moving to the left/down
velocity is negative
uv' + vu'
no limits - find antiderivative + C - use inital value to find C
use rectangles with right-endpoints to evaluate integrals (estimate area)
32. methods of integration
if integral converges - series converges
v(dx/dt)² + (dy/dt)² not an integral!
velocity is positive
substitution - parts - partial fractions
33. definite integral
has limits a & b - find antiderivative - F(b) - F(a)
(uv'-vu')/v²
draw short segments representing slope at each point
critical points and endpoints
34. Alternating series tes
lim as n approaches zero of general term = 0 and terms decrease - series converges
Slope of secant line between two points - use to estimate instantanous rate of change at a point.
uv' + vu'
? A(x) dx over interval a to b - where A(x) is the area of the given cross-section in terms of x
35. 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 = 1/n^p - converges if p > 1
no limits - find antiderivative + C - use inital value to find C
f(x) has a relative minimum
36. Indeterminate forms
decreasing
lim as n approaches 8 of ratio of (n+1) term/nth term > 1 - series converges
positive
0/0 - 8/8 - 8*0 - 8 - 8 - 1^8 - 0° - 8°
37. dP/dt = kP(M - P)
logistic differential equation - M = carrying capacity
? 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
y' = -csc²(x)
38. Alternate definition of derivative
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 over interval from a to b
general term = a1r^n - converges if -1 < r < 1
Limit as x approaches a of [f(x)-f(a)]/(x-a)
39. Instantenous Rate of Change
velocity is positive
positive
Slope of tangent line at a point - value of derivative at a point
0/0 - 8/8 - 8*0 - 8 - 8 - 1^8 - 0° - 8°
40. Volume of solid with base in the plane and given cross-section
? A(x) dx over interval a to b - where A(x) is the area of the given cross-section in terms of x
uv - ? v du
use rectangles with left-endpoints to evaluate integral (estimate area)
A function and it's derivative are in the integrand
41. nth term test
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)
if terms grow without bound - series diverges
y' = 1/(1 + x²)
y' = -1/(1 + x²)
42. When f '(x) changes from negative to positive - f(x) has a...
polynomial with infinite number of terms - includes general term
relative minimum
1/2 ? R² - r² over interval from a to b - find a & b by setting equations equal - solve for theta.
negative
43. rate
use tangent line to approximate values of the function
product rule
zero
derivative
44. Given velocity vectors dx/dt and dy/dt - find total distance travelled
f(x) has a relative minimum
? v (dx/dt)² + (dy/dt)² over interval from a to b
uv - ? v du
if lim as n approaches 8 of ratio of comparison series/general term is positive and finite - then series behaves like comparison series
45. y = e^x - y' = y' =
p ? r² dx over interval a to b - where r = distance from curve to axis of revolution
1/(b-a) ? f(x) dx on interval a to b
? v(1 + (dy/dx)²) dx over interval a to b
e^x
46. y = x cos(x) - state rule used to find derivative
product rule
(uv'-vu')/v²
Limit as x approaches a of [f(x)-f(a)]/(x-a)
draw short segments representing slope at each point
47. Find interval of convergence
y' = sec(x)tan(x)
p ? r² dx over interval a to b - where r = distance from curve to axis of revolution
? v(1 + (dy/dx)²) dx over interval a to b
use ratio test - set > 1 and solve absolute value equations - check endpoints
48. 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
relative minimum
lim as n approaches zero of general term = 0 and terms decrease - series converges
y' = 1/x
49. When f '(x) is increasing - f(x) is...
if integral converges - series converges
(uv'-vu')/v²
concave up
y' = -csc²(x)
50. y = log (base a) x - y' =