Brainscape's Knowledge GenomeTM

Browse over 1 million classes created by top students, professors, publishers, and experts.


See full index

AP Calculus BC > Midterm - Terms to Know > Flashcards

Midterm - Terms to Know Flashcards

1
Q

instantaneous velocity

A

lim(h–>0) of [f(x1 + h) - f(x)]/h

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
2
Q

lim(x–>0) of sinx/x

A

1

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
3
Q

lim(x–>inf.) of sinx/x

A

0

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
4
Q

lim(x–>c) of A

A

A

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
5
Q

lim(x–>c) of x

A

c

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
6
Q

lim(x–>c) of [f(x) +/- g(x)]

A

lim(x–>c) of f(x) +/- lim(x–>c) of g(x)

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
7
Q

lim(x–>c) of [kg(x)]

A

k*lim(x–>c) of g(x)

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
8
Q

lim(x–>c) of [f(x)^n]

A

[lim(x–>c) of f(x)]^n

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
9
Q

Sandwich Theorem

A

if g(x) </= f(x) </= h(x) in some interval about c, and lim(x–>c) of g(x) = lim(x–>c) of h(x), then lim(x–>c) of f(x) = L

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
10
Q

a function is continuous if…

A

the value exists
the limit exists
the limit equals the value

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
11
Q

intermediate value theorem

A

if a function is continuous between points a and b, it will take on every value between f(a) and f(b)

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
12
Q

tangent line equation

A

ytan = m(x-x1) + y1

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
13
Q

what is a normal line?

A

a line perpendicular to the tangent line; has the opposite reciprocal slope

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
14
Q

when is a function differentiable?

A

if it has a derivative at every point in its domain, and is both continuous and smooth

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
15
Q

power rule

A

d/dx(x^n) = nx^(n-1)

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
16
Q

constant multiple rule

A

d/dx(cu) = c*d/dx(u)

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
17
Q

sum and difference rule

A

d/dx(u +/- v) = du/dx +/- dv/dx

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
18
Q

product rule

A

d/dx(uv) = du/dxv + dv/dxu

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
19
Q

quotient rule

A

“low d high minus high d low, square the bottom and away we go”
d/dx(u/v) = (du/dxv - dv/dxu)/v^2

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
20
Q

position, velocity, acceleration relationships

A

velocity is the first derivative of position
acceleration is the first derivative of velocity and second derivative of position

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
21
Q

velocity and acceleration

A

when velocity is + or - but increasing, acceleration is +
when velocity is constant, acceleration is 0
when velocity is decreasing, acceleration is -

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
22
Q

d/dx(sinx)

A

cos(x)

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
23
Q

d/dx(cosx)

A

-sin(x)

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
24
Q

d/dx(tanx)

A

sec^2(x)

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
25
Q

d/dx(secx)

A

secxtanx

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
26
Q

d/dx(cscx)

A

-cscxcotx

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
27
Q

d/dx(cotx)

A

-csc^2x

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
28
Q

chain rule

A

dy/dx = dy/du * du/dx
y’ = f’(g(x)) * g’(x)

29
Q

inverse functions

A

derivative of a function’s inverse is the reciprocal of that function’s derivative

30
Q

d/dx(sin-1x)

A

1/(rt[1-x^2])

31
Q

d/dx(cos-1x)

A

-1/(rt[1-x^2])

32
Q

d/dx(tan-1x)

A

1/(1+x^2)

33
Q

d/dx(cot-1x)

A

-1/(1+x^2)

34
Q

d/dx(secx)

A

1/(|u|*rt[u^2-1])

35
Q

d/dx(cscx)

A

-1/(|u|*rt[u^2-1])

36
Q

d/dx(e^x)

A

e^x

37
Q

d/dx(lnx)

A

1/x

38
Q

extreme value theorem

A

if f is continuous over a closed interval, then f also has a maximum and a minimum value over that closed interval, including at its endpoints
–> maximum/minimum are where f’ = 0

39
Q

first derivative test

A

extrema exist where f exists and f’ changes signs (+ –> - or - –> +)
neg. to pos.: minimum
pos. to neg.: maximum

40
Q

mean value theorem

A

if f(x) is continuous over [a, b] and differentiable over (a, b), then at some point c between a and b, f’(c) = (f(b)-f(a))/(b-a)

41
Q

increasing/decreasing functions

A

increasing if: when x1 < x2, f(x1) < f(x2)
decreasing if: when x1 < x2, f(x1) > f(x2)

42
Q

antipower rule

A

if f(x) = ax^n, F(x) = (a/([n+1])x^(n+1)

43
Q

curve sketching

A

concave up: when f ‘’ (x) > 0
concave down: when f ‘’ (x) < 0

concave up/increasing: f’ > 0, f ‘’ > 0
concave up/decreasing: f’ < 0, f ‘’ > 0
concave down/increasing: f’ > 0, f ‘’ < 0
concave down/decreasing: f’ < 0, f ‘’ < 0

44
Q

related rates tip

A

find the rate at which one quantity is changing by relating it to other quantities, whose rates of change are known
1. draw a picture
2. write down what you know and what you’re looking for
3. write an equation relating knowns and unknowns
4. take derivative, usually with respect to time (t)
5. solve (by plugging in what you know) and interpret the answer

45
Q

rectangle/square formulas

A

P = 2l + 2w
A = lw

46
Q

parallelogram formulas

A

P = 2a + 2b
A = bh (h = height, not side length)

47
Q

triangle formulas

A

P = a + b + c
A = (1/2)bh

48
Q

trapezoid formulas

A

P = a1 + a2 + b1 + b2
A = (1/2)h(b1 + b2)

49
Q

circle formulas

A

P = 2pir
A = pir^2

50
Q

rectangular prism formulas

A

V = lwh
SA = 2lw + 2lh + 2wh

51
Q

cone formulas

A

V = (1/3)pir^2h
SA = pir^2 + pi*rs (s = length of side)

52
Q

sphere formulas

A

V = (4/3)pir^3
SA = 4pir^2

53
Q

pyramid formulas

A

V = (1/3)(ab)h
SA = (1/3)(2a + 2b)(s)(ab)
where s = height of triangle face

54
Q

cylinder formulas

A

V = pir^2h
SA = 2pir^2 + 2pi*rh

55
Q

linearization of f at a

A

L(x) = f(a) + f’(a)(x-a)

56
Q

Newton’s method

A

used to find approximate roots of equations using linearizations
1. find function & derivative –> take a guess at a root, then find the linearization of the curve at that point, where L(x) = 0
2. use x-value found from linearization to guess again; x(n+1) = xn - (f[xn]/f’[xn])
3. find where x stops changing

57
Q

optimization steps

A
  1. draw scenario
  2. write equation for quantity you want to optimize in terms of one variable
  3. find first derivative of equation and set equal to 0
  4. check endpoints if necessary
58
Q

approximation methods for integrals

A

LRAM: A = h(y0 + y1 + y2 + y3) where y-values are from left corner
RRAM: A = h(y1 + y2 + y3 + y4) where y-values are from right corner
MRAM: A = h(y0.5 + y1.5 + y2.5 + y3.5)

59
Q

calculating area under the curve - definite integral notation

A

lim(n–>inf.) of the sum of all f(ck)(delta of x) = intg. from b –> a of (f(x)dx)

60
Q

integral evaluation theorem

A

intg. from b–>a of (f(x)*dx) = F(b) - F(a)

61
Q

rules for integrals

A

reversing limit changes the sign (+/-)
if upper and lower limits are equal, area = 0
constant multiples can be moved outside
integrals can be added/subtracted
intervals can be added/subtracted
neither integrals nor intervals can be multipled/divided

62
Q

mean value theorem for intervals

A

1/(b-a) * intg. from b–>a of (f(x)*dx)

63
Q

fundamental theorem

A

d/dx of (intg. from x–>a of f(t)*dt) = f(x)
need parameters:
- derivative of integral
- derivative matches upper limit of integration (d/dx matches x)
- lower limit is a constant (a)
*integral can be rearranged to suit these parameters

64
Q

trapezoidal rule

A

T = (h/2)(y0 + 2y1 + 2y2…+2y(n-1) + yn)

65
Q

Simpson’s Rule

A

T = (h/3)(y0 + 4y1 + 2y2 + 4y3…+2y(n-2) + 4y(n-1) + yn)

66
Q

integration by parts formula

A

intg. of (udv) = uv - intg. (vdu)
u must differentiate to 0, dv must be easy to integrate
*choose u using LIPET

67
Q

tabular integration works for…

A

integrals of the form intg. (f(x) * g(x)*dx), where f(x) differentiates to 0 and g(x) integrates forever

68
Q

slope fields

A

if x is affecting, change horizontally
if y is affecting, change vertically

AP Calculus BC (4 decks)

Brainscape helps you reach your goals faster, through stronger study habits.
© 2024 Bold Learning Solutions. Terms and Conditions