Curve Sketching and More Derivatives

d/dx e^x =

e^x

d/dx e^u =

u’ * e^u

d/dx a^x =

a^x * lna

d/dx a^u =

a^u * lna * u’

d/dx lnx =

1 / x

d/dx lnu =

u’ / u

d/dx log_ax =

1 / (x * lna)

d/dx log_au =

u’ / (u * lna)

d/dx arcsinu =

u’ / √(1-u²)

d/dx arccosu =

-u’ / √(1-u²)

d/dx arctanu =

u’ / (u² + 1)

d/dx arccotu =

-u’ / (u² + 1)

d/dx arcsecu =

u’ / |u|√(u² - 1)

d/dx arccscu =

-u’ / |u|√(u² - 1)

d/dx f^-1(x) =

1 / f’(f^-1(x))

Antiderivative

Original function from a derivative with a +c at the end

What does this mean:

For all x, f(g(x)) = x and g(f(x)) = x

g(x) = f^-1(x)

The functions are inverses of each other

L’Hopital’s Rule

if lim(x→c) f(x) / g(x) = 0/0 or ∞/∞, then lim(x→c) f(x) / g(x) =

lim(x→c) f’(x) / g’(x)

If L’Hopital’s rule can’t be used, then the limit…

does not exist

Steps to use L’Hopital’s Rule

1. Take derivative of numerator and denominator of original function in the limit

2. Simplify and plug in c value that x approaches in the limit

3. Solve and if 0 is in denominator then the limit doesn’t exist

Extema

Max and min points

Specificity of extrema from least to most specific

Critical point, local max/min, absolute max/min

Critical point

An x-value within the domain of a function where the derivative is zero or undefined (general descriptor)

Extreme value theorem

If f is continuous on [a, b], then f has both a max and min value on the interval

Can a critical point be a number that’s undefined in the domain

No

How to find exact absolute max and min values of a function on a given interval

1. Find derivative of function

2. Find critical points by setting derivative equal to 0

3. Plug critical points and endpoints on the interval back into the original function

4. See which values are highest and lowest

How to find when the derivative is undefined

Set denominator equal to 0 and check log functions

How to use the 1st derivative test

1. Find critical values and create intervals

2. Test values within each interval in f’(x)

3. See if pos or neg because pos indicates that f(x) is increasing and neg indicates decreasing

4. Analyze trends between increasing and decreasing between critical points to find extrema

Sign chart: increasing to decreasing =

max

Sign chart: decreasing to increasing =

min

Extrema occur at ____ ____ in the derivative

sign changes

In the graph of a derivative, a zero without a sign change in the graph (bounces off x-axis) indicates…

a critical point but not an extreme value

Point of inflection

Change in concavity on f when…

• f’’(x) = 0 or DNE

• f’’(x) changes signs

• When f’(x) changes from increasing to decreasing or vice versa (same thing as f’’(x) changing signs)

2nd derivative test: f’’(c) > 0, then…

f is concave up and has a relative min at x = c

2nd derivative test: f’’(c) < 0, then…

f is concave down has a relative max at x = c

If f’’(c) = 0, then the test is…

inconclusive

Critical points are not…

concavity changes

In the graph of f’, a bracket is used in concavity intervals for…

defined endpoints of the graph (filled in circle)

Positive f’’(x) indicates that the graph is…

concave up

Negative f’’(x) indicates that the graph is…

concave down

How to use the 2nd derivative test to find relative extrema

1. Take derivative of function and find critical points where f’(x) = 0

2. Find the 2nd derivate and plug in critical points to test positivity (what the sign is- pos or neg)

3. Negative output indicates concave down and a max, and opposite for positive

A critical point is the same as a…

turning point

Critical points in the graph of f’ can occur at x-intercepts and…

holes/discontinuities

(because crit pts occur when f’ is 0 or undefined)

The 2nd derivative test can only be used for critical points where f’(c) =

0

NOT UNDEFINED CRIT PTS

Even if f’’(x) = 0 at some x value, it could NOT be a point of inflection if…

f’’(x) doesn’t change signs from pos to neg or neg to pos

Do POIs occur when the graph of f’ is flat?

No (no concavity change)

How to find intervals of concavity

1. Find POIs by setting f’’(x) = 0

2. Test values in f’’(x) around the POI intervals

3. Positive = conc up, negative = conc down

A particle is farthest left on x(t) at the….

relative minimum

A particle is farthest left on v(t) at the….

x-intersection where the graph goes from negative to positive

(at this point, the particle has traveled the longest left even though it’s slowing down)

How to find when the velocity of the particle is increasing the fastest

Find the relative max of a(t) by finding a’(t)

Endpoints are only critical points when…

they’re an absolute max/min

f and g are inverses. f(1) = 5

Critical points exist at x-values where the derivative equals zero or is…

undefined (don’t forget to check undefined values!!!!!)

Use the first derivative test and sign chart to find…

relative extrema based on increasing and decreasing changes from f’(x) sign changes

Can e^u ever equal 0

No