unit 4 Flashcards
(28 cards)
sinx
cosx
cosx
-sinx
tanx
sex^2x
how do u derive
- derive outer function
- multiply by dervative of inner function
- bring inner derivative to front
slope of tangent line on graph given x-value
- derive given equation
- plug in given x-value in derived eqn
- use special triangles to get exact value nd rationalize
equation of tangent line on graph given an x-value
- find y-value by plugging in given x
- find slope (m) by deriving orig eqn nd plugging in x-val
- plug y, x, and m into y=mx+b to isolate for b (yes, it may have many terms)
- make final eqn
properties of e
- irrational number like pi
- log(e)x is natural logarithm nd writtn as lnx
- ** log(e)e=lne=1 **
how do u graph e^x
plug in -1 to 1 in x
how do u graph y=lnx
- x-vals are y-vals from e^x graph
- plug into calc
how r the graphs of lnx and e^x related?
- inverses of each other
- “reflected” across y=x line
- asymptotes at x=0 and y=0
whts the value of lne
1
new rule in exponential functions
if f(x)=b^x
then f’(x)=b^xlnb
- orig multiplied by ln(base)
determine number of ___ after # of weeks
- plug values into P(t)=P(o)(increase)^time
- calculate
how fast is the population increasing?
- derive original equation without a time value (in exponent)
- plug in time and calculate
- “increasing by approximately ___ insects per week”
determine initial population (given formula)
- put variable e right so that exponent is applied to it
- plug in x=0 (ignore e)
how long will it take for pop’n to double?
- multiply initial pop’n by 2
- set equation (ith e) to doubled pop’n val
- isolate for t by multiplying ln on both sides
rewrite a population function as an exponential function w/ base 2
- plug into A(t)=A(o)(2)^(t/D)
*D is the doubling period, calculated previously
*no value is plugged in for t
wht does it mean when there’s x in two locations when deriving function?
- cant treat x as constant
- must use product rule/quotient rule
derive y=2e^x cosx
- use product rule bc 2e^x and cosx are two functions
- keep 2 as constant coeff at front
identify local extrema for exponential function
- derive
- factor and set to zero
MUST SAY 0=e^x doesn’t work bc y doesn’t reach 0 on graph - plug in each x-val to the original function to find max/min vals
find max/min in word problems
- derive
- set to zero (**if u have e to the power of something, must state tht it is unuseable based on graph)
- plug x-vals (as well as from range) into original equation the CHECK tht we do have the highest y-val
lambda?
- disintegration constant
- how fast radioactive decay occurs
determine disintegration constant
- plug in given values (t, F(t), N(o))
- isolate for lambda by taking ln of both sides
determine half-life
- divide initial concntration by 2 to get N(t)
- set whole eqn to that half value + make sure to put e right before the exponent
- take ln of both sides, isolate for time
