Math 142

d/dx (x^n)

nx^n-1

d/dx (b^x)

(b^x)ln(b)

d/dx e^x

e^x

d/dx (logb(x))

1/(xln(b))

d/dx (ln(x))

1/x

How do you find exact profit?

y (46)-y(45)

How do you estimate profit?

P' (45)

What is the product rule?

y=f(x)*g’(x)+g(x)*f’(x)

What is the quotient rule?

g(x)*f’(x)-f(x)*g’(x)/(g(x))²

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

f’(g(x))*g’(x)

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)

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

increasing; >0

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

decreasing; <0

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

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

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.

Estimate of a single item

C'(n-1)

Exact/marginal cost of a single item

C(n)-C(n-1)