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Ap Calc AB > Cram deck > Flashcards

Cram deck Flashcards

1
Q

What is the derivative of a constant

A

Zero

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2
Q

what is the derivative of (k x f(x))

A

(k)f’(x)

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3
Q

what is the derivative of (f(x) + g(x))

A

f’(x) + g’(x)

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4
Q

what is the derivative of (f(x) x g(x))

A

f(x)g’(x) + f’(x)g(x)

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5
Q

what is the derivative of f(x)/g(x)

A

g(x)f’(x)- g’(x)f(x)/ g(x)^2

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6
Q

what is the derivtive of sin(f(x)

A

cos(f(x)) x f’(x)

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7
Q

what is the derivative of cos(f(x)

A

-sin(f(x) x f’(x)

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8
Q

what is the derivative of tan(f(x)

A

sec^2 (f(x) x f’(x)

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9
Q

what is the derivative of ln(f(x)

A

1/f(x) x f’(x)

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10
Q

what is the derivative of e^f(x)

A

e^f(x) x f’(x)

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11
Q

what is the derivative of a^f(x)

A

a^f(x) x lna x f’(x)

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12
Q

what is the derivative of sin^-1 f(x)

A

f’(x)/ (1-f(x)^2)^.5

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13
Q

what is the derivative of cos^-1 f(x)

A

-f’(x)/ (1 - f(x)^2)^.5

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14
Q

what is the derivative of tan^-1 f(x)

A

f’(x)/ (1+ f(x)^2)

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15
Q

what is the derivative of f^-1 (x) at x= f(a)

A

1/ f’(x) at x=a

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16
Q

what is the L’hopitals rule:

A

If lim x>a f(x)/g(x) = 0/0 and of lim x>a f’(x)/g’(x) exsists then Iim x>a f(x)/g(x) = f’(x)/g’(x)
the same applies for when x> infinity of for a infinity over infinity form

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17
Q

what is a critical point

A

any c in the domain of f such that either f’(c) =0 or f’(c) = und. This is called a critical point or critical value of f

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18
Q

what is the equation of the tan line to the curve y=f(x) at x=a

A

y - f(a) = f’(a)(x-a)

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19
Q

what can the tangent line to the graph be used for

A

can be used to approximate a function value at points very near the point of tangency. This is known as linear approximation. make sure to use the approximation symbol instead of =

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20
Q

what is the equation of the normal line (perpendicular) to the curve f(x) at x=a

A

y - f(a) = -1/f’(a) x (x-a)

21
Q

how do you know if a function is increasing/decreasing

A

if the derivative is positive it is increasing, if its negative its decreasing

22
Q

how do you know if the curve y=f(x) has a local(relative) minimum at a point where x=c

A

if the first derivative changes signs from negative to positive at x=c

23
Q

how do you know if the curve y=f(x) has a local(relative) maximum at a point where x=c

A

if the first derivative changes signs from positive to negative at x=c

24
Q

how do you know if y=f(x) is concave upward

A

if the second derivative is positive on that interval. if it is negative on the interval than it is concave downward

25
Q

what is the point of inflection

A

where the concavity of y=f(x) changes

26
Q

how do you know if y=f(x) has an absolute minimum at x=c on closed interval (a,b)

A

if f(c) is less than all other values on that interval. it is a maximum if f(c) id greater than all y values on (a,b)

27
Q

related rates

A

if several functions of time t are related through an equation such as Pythagorean theorem then we can obtain a relation involving their (time)rates of change by differentiating with respect to x

28
Q

approximating areas

A

you can approximate the value of a definite integral. If f is nonnegative on (a,b) then we interpret the area as the region bounded by y=f(x), the x-axis, and lines x=a and x=b. the value of the integrated is approximated by dividing the area up into n number of strips, approximating the area of each strip with a rectangle or other geometric figures, then summing these approximations.

29
Q

what is the difference between a left sum, right sum, midpoint sum, and a trapaziodal approximation

A

on the left sum, you start at the left and don’t include the far right. for the right sum, you start right and don’t include the left endpoint. for middle you take the average values of the two-point of delta x. for these 3 you multiple delta x by f(0) + f(1)…. for trapezoidal approximation you multiply by delta x/2 and all of the f(x) values being summed are doubled except for the first and last ones.

30
Q

what is the integral of k x f(x)dx

A

k x (integral of) f(x)dx

31
Q

what is the integral of f(x) + or - g(x) dx

A

(integral) f(x)dx + or - (integral) g(x)dx

32
Q

(integral) u^n du =?

A

u^(n+1)/(n+1) +c

33
Q

(integral) 1/n du

A

ln (u) +c

34
Q

(integral) cos u du

A

sin u +c

35
Q

(integral) sin u du

A

-cos u +c

36
Q

(integral) tan u du

A

ln (sec u) +c

37
Q

(integral) sec^2 u du

A

tan u +c

38
Q

(integral) e^u du

A

e^u +c

39
Q

(integral) a^u du

A

a^n/ lna +c

40
Q

(integral) 1/ (a^2 - u^2)^.5 du

A

sin^-1 u/a +c

41
Q

(integral) 1/(a^2 + u^2)

A

1/a tan^-1 u/a +c

42
Q

The first fundamental theorem of calculus

A

if if is continuous on closed interval (a,b) and f’=f, then

integral) from a to b = F(b) - F(a

43
Q

the second fundamental theorem of calculus

A

if f is continuous on (a,b) then the function F(x)= (integral) from a to x f(t) dt has a derivative at every point in (a,b) and F’(x) = d/dx (integral) from a to x of f(t) dt = f(x)

44
Q

(integral) from a to a f(x) =

A

0

45
Q

(integral) from a to b f(x) dx is flipped

A
  • (integral) from b to a f(x) dx
46
Q

(integral) from a to b f(x) dx is split up

A

(integral) from a to c f(x) dx + (integral) from c to b f(x) dx

47
Q

what is the area under the curve if there are negative y values

A

area above - area below.

48
Q

the volume of a solid of revolution (consisting of disks is given by

A

Volume= pi (integral) from a to b r^2 dx or dy

49
Q

the volume of a solid of revolution (consisting of washers) is given by

A

Volume = pi (integral) from a to b outside R^2 - inside r^2 dx or dy

Ap Calc AB (1 decks)

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