Math 142

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d/dx (x^n)

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Calculus

18 Terms

1

d/dx (x^n)

nx^n-1

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2

d/dx (b^x)

(b^x)ln(b)

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3

d/dx e^x

e^x

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4

d/dx (logb(x))

1/(xln(b))

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5

d/dx (ln(x))

1/x

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6

How do you find exact profit?

y (46)-y(45)

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7

How do you estimate profit?

P' (45)

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8

What is the product rule?

y=f(x)*g’(x)+g(x)*f’(x)

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9

What is the quotient rule?

g(x)*f’(x)-f(x)*g’(x)/(g(x))²

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10

d/dx (f(g(x))) =

f’(g(x))*g’(x)

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11

How do you find where intervals are increasing or decreasing

  1. Domain of f

  2. Partition numbers of f’ (f’=0 and f’ DNE)

  3. Critical values (partition values in the domain of f)

  4. Sign chart of f’(x)

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12

if f is concave up, then f’ is ________ and f” is _____

increasing; >0

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13

if f is concave down, then f’ is _________ and f” is______

decreasing; <0

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14

How do you find local maxima

  1. determine the domain of f

  2. What are the critical values of f

  3. Find f” and evaluate it where f’=0 then determine whether f has local extrema

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15

What is the second derivative test

  • If f”<0 then f is concave down and has a local maximum at x=c

  • If f”>0 then f is concave up and has a local minimum at x=c

  • If f”=0 then the test fails and f may have a local maximum, minimum, or neither at x=c

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16

Graph f(x) without technology

  1. find the domain, x- and y-intercepts, holes, and any vertical or horizontal asymptotes of f.

  2. Find the intervals where f is increasing/ decreasing and any local extrema using the First Derivative Test

  3. Find theintervals of concavity and any inflection points of f using the Concavity Test

  4. Create a shape chart for f by combining the sign charts from steps 1 & 2 and draw ing the resulting "shape" in each interval

  5. Create a rough sketch of f using the shapes from step 4 and the information from step 1.

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17

Estimate of a single item

C'(n-1)

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18

Exact/marginal cost of a single item

C(n)-C(n-1)

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