Calc Final Exam

How can a function fail to differentiable

sharp corner, a discontinuity (lim from left and right don't match), vertical tangent

if there is a hole in a line

it limit exists but it is not continuous

lnx graph

e^x graph

quotient rule

lo x Dhi - hi x Dlo/ lolo

derivative sinx

cosx

derivative cosx

-sinx

derivative tanx

sec^2x

derivative cscx

-cscxcotx

derivative secx

secxtanx

derivative cotx

-csc^2x

cscx

1/sinx

secx

1/cosx

cotx

cosx/sinx

tanx

sinx/cosx

lim (x approach 0) sinx/x =

1

lim (x approach 0) cosx-1/x=

0

5^2x

5^2x · ln5 · 2. Original · ln(outerfunc)· derivative inner function

implicit differentiation

dy/dx for ever y derivative and solve dy/dx

tan^-1 x

1/1+x^2

area of a circle

π r^2

circumference

2 π r

definition of the derivative

f(x+h)-f(x)/ h

linearization formula

f(a)+f '(a)(x-a)

mean value theorem

f(b)-f(a)/ b-a= c

mean value theorem graph

find c using f(b)-f(a)/ b-a= c draw tan lin

antiderivative sinx

-cosx + C

antiderivative cosx

sinx + C

antiderivative sec^2 c=x

tanx + c

antiderivative secxtanx

secx+c

antiderivative csc^2 x

-cotx+c

cscxcotx

-cscx+c

1/x

ln|x| + c

Linear approximation formula (first step)

f(a) +f'(a)(x-a)

Linear approximation estimate

separate how it differs from original (10^2 v 10.01^2= (10+0.01)^2 when plug in the 0.01 for x from linear approx formula

find the differentiation

solve dy

Δy

f(Δx+x) - f(x)

Find exact values of each expression without calculator.

Either make the number able to cancel out (log9(9^2)), Cancel out ln and e, or expand out to simplify

Use law of logarithm to expand each expression

Use law of logs to expand OUT.

Express a simple logarithm

use rule to make it smaller with ONE ln or log

x/0

dne

Show function is continuous

1. f(x) is defined
2. Limit as f(x) approaches x exists (MUST MATCH ON BOTH SIDES, right hand limit and left hand limit)
3. f(x) = limit
4. STATE CONCLUSION IS CONT OR NOT

Intermediate Value Thm to show solution

1. let function equal 0, explain continuous based of polynomial etc
2. Pick n which falls in domain( it can equal one), plug in range into equation and prove that n falls between
3. By ivt # c exists in (0,1)

lim x approachs a. formula

f(x)- f(a)/x-a

find f’a

plug in a instead of x into lim h-0 (f(a+h)-f(a)/h)

Abs max and min

plug critical # and domain end points into original function

Rolle’s thm

1. conti
2. diff
3. equal

STATE CONCLUSION

Indeterminate forms

\n ∞-∞, 0/0, ∞/∞, 0^0, ∞^0, 1^∞

b^x, antiderivative

b^x/lnb + C

Use midpoint rule with n = 4 approximate integral

take range \[0,2\] and divide absolute value by n, THAN plug in midpoints into formula added together and multiply by rectangle width

If you are taking the derivative integral function and x is not right form

let x= u proceed as normal and multiply by x’

evaluate integral

take inner function | with upper and lower limits and plug in upper - lower

find general indefinite integral

derivative PLUS C

find definite integral

derivative upper limit- lower limit plugged in

express the limit as a definite integral

have the squiggly thing with upper and lower limit, take i out, and MULITIPLY by dx