calc 1 unit 2 test
m^3 − n^3 =
(m − n)(m2 + mn + n 2 )
arbitrary integer
+period and n, used when no interval for trig functions
critical numbers =0,18/11, and 3. the second derative is 30x^5(x-3)
- can only use the second deterative test for 18/11 to see if its a max or min - will have to use 1st derative test to find all max and min values
Find a cubic function f(x) = ax3 + bx2 + cx + d that has a local maximum value of 4 at x = −2 and a local minimum value of 0 at x = 1.
- can plug in x values - can take f'(x) at x values, euqla it to zero -system of equations
Find the points where the curve y=∣x+ 2)/(x^2+ 5)∣ has horizontal tangents.
set function to different ones, solve for each differvaite then set each one to 0 answer: (1,1/2) (-5,1/10)
The quantity (in pounds) of a gourmet ground coffee that is sold by a coffee company at a price of p dollars per pound is Q = f(p).
The rate of change of the quantity of coffee sold with respect to the price per pound when the price is $8 per pound. units: pounds/(dollar/lb)). f ′ (8) is negative since the quantity of coffee sold will decrease as the price charged for it increases. People are generally less willing to buy a product when its price increases.
where are there points of tangency to y=(1-2x)^2 that pass through (1,-3)
x=a, y=(1-2a)^2 y'(a), x=1, x=-3, plug into point slope and solve for a a=0,2 answer: (0,1) and (2,9)
Mean Value Theorem
if f(x) is continuous and differentiable, slope of tangent line equals slope of secant line at least once in the interval (a, b) f '(c) = [f(b) - f(a)]/(b - a)
orthogonal
if their tangent lines are perpendicular at each point of intersection. solve to see if their derivatives are perpendicular
when faced with a tricky product rule
remeber u can factor out terms to solve for zero
tanx
sec^2x
on mean value theorm problems
secent line and tangent line are parallel because the slope of the secent line is point slope and so it tangent
secx
secxtanx
Use L(x) to approximate the numbers 0.9^1/2 and 0.99^1/2 f(x)=(1-x)^1/2 l(x)=1-x/2
set .9^1/2 equal to f(x) to find x and then plug x into l(x). .9^1/2=.9500 .99^1/2=.9950
limh->pi/3 cos(x)-1/2/x-pi/3
f(x)=cos(x) f'(pi/3)=-3^1/2/2
how is differential same thing as linarization
f(x)=l(x) f(x)=f(a) +f'(a)(x-a) delta f=f'(a)(x-a) this is because dy=f'(a)(x-a) do dy=f'(x)dx
when it says find the local maxium and min values
find it using the 1st and 2nd derative test. make sure to label the tests and explain what ur doing and why
increasing and decreasing test min and max wise
if f' changes from + to - then there is a local max - to + then there is local min at c if it doesn't change then there is no max or min
when given a constant
plug into equation and will be treated a cinstant, either be 0 if added or constant if added
the average rate of change ofC
point slope of interval, can't check with derative
s(t) =
position (net displacement)
vertical tangents
possible when y' is undefined
increasing and decreasing test
(a) If f'(x) > 0 on an interval, then f is increasing on that interval. (b) If f'(x) < 0 on an interval, then f is decreasing on that interval.
cotx
-csc^2x
cscx
-cscxcotx
dy/dx
-inline with chain rule -don't know how to take y's derivative directly
d/dx cosx
-sinx
in order to get credit
-when taking derative, make sure you do d/dx -write out eveyrthing, espeically in related rates -
related rates
1. Draw a diagram. 2. Assign variables to all quantities that are functions of time. 3. Express the given information in terms of derivatives. 4. Write an equation that relates the various quantities. 5. Differentiate the equation with respect to time using the chain rule. 6. Substitute the given values and solve for the unknown rate.
conditions for differentiability
1. continuous 2. kinks 3. vertical tangent line
remember with related rates
1. reduce all variables if possible, u cant have two at once so you may have to sub one in for another. remember this !!! 2.
prove that x^3/3 - cos(x) +2x has exactly one root
1. use IVT to show theres atleast one root 2. cont? yes f(0) = -1 f(2) = >0 proven there is atleast one 2. show there can't be a second one 2 roots , f(a)=0, f(b)=0 so f(a)=f(b) 3. prove that f'a cant be equal to zero
Volume of a cone
1/3πr²h
Find an equation of the normal line to the parabola y = x2 − 9x + 7 that is parallel to the line x − 5y = 5.
1/5x-27/5=y -1/m = the deterative of the parabola
At noon, ship A is 90 km west of ship B. Ship A is sailing south at 25 km/h and ship B is sailing north at 5 km/h. How fast is the distance between the ships changing at 4:00 PM? (Round your answer to one decimal place.)
24 km/h
Circumference of a circle
2πr
Surface Area of a Cylinder
2πrh+2πr²
Volume of a sphere
4/3πr³
Surface area of a sphere
4πr²
as a particle moves along curve y=x^2 as the particle moves through (1,1) the x coordinate increases by 2 cm/s. how fast is the distance to the origin changing?
6 cm/ sqaure root of 2 cm/s
v(t)
= velocity (direction/sign and mag)
"Veritcal Asymptote"
A line x = a is a vertical asymptoteif lim x→a− f(x) = ±∞ or lim x→a+ f(x) = ±∞
horizontal asymptotes
A line y = L is a horizontal asymptote if limx→∞ f(x) = L or lim x→−∞ f(x) = L.
Tangent to a curve
a straight line that touches but does not cut the curve at a particular point
Area of a circle
A=πr²
If f '(3) = 0 and f ''(3) = 5, what can you say aboutf?
At x = 3, f has a local minimum.
If f '(5) = 0 and f ''(5) = 0, what can you say aboutf?
At x = 5, f has neither a maximum nor a minimum.
Fermat's Theorem
If f has a local maximum or minimum value at c, and if f'(c) exists, then f' (c) = 0.
Extreme Value Theorem
If f is continuous on [a,b] then f has an absolute maximum and an absolute minimum on [a,b]. The global extrema occur at critical points in the interval or at endpoints of the interval.
Intermediate Value Theorem (IVT)
If f is continuous on the closed interval [a,b] and k is any number between f(a) and f(b), then there is at least one number c in [a,b] such that f(c)=k
Concavity Test
If f''(x) > 0 for all x in I, then the graph of f is concave upward on I. If f''(x) < 0 for all x in I, then the graph of f is concave downward on I.
2nd derivative test for local extrema
If f'(c) = 0 and f"(c)<0, there is a local max on f at x=c. If f'(c) = 0 and f"(c)>0, there is a local min on f at x=c.
where l(x) comes from
L(x)-f(a)= f'(a) (x-a) y-y1=m(x-x1) a and f(a) is tangent line
rolles theorem
Let f be continuous on [a,b] and differentiable on (a,b) and if f(a)=f(b) then there is at least one number c on (a,b) such that f'(c)=0 (If the slope of the secant is 0, the derivative must = 0 somewhere in the interval).
isosceles triangles
Most popular is the 90 degree: 45-45-90
Show that y = x 1/3 has a vertical tangent line at (0, 0).
Note that f(x) is continuous at x= 0 (since it is a root function) and from part (a) we see that limx→0 |f′(x)|= limx→0 (1/3x^2/3=∞ and so there is a vertical tangent at x= 0.
relative error and percent error
Relative: dv/v percent: dv/v times 100 write this work out, not in numbers, chekc ur answer with the numbers tho
horizontal tangents
Set numerator of derivative equal to 0. can also be solved for with implicit differentiation
Find all values c in (−27, 27) such that f '(c) = 0. (Enter your answers as a comma-separated list. c=DNE why doesnt this contradict rolles theorm
This does not contradict Rolle's Theorem, since f '(0) does not exist, and so f is not differentiable on (−27, 27).
first derivative test for local extrema
Used to determine where a function's graph has a min/max and is increasing or decreasing.
Volume of a cylinder
V=πr²h
For what values of a and b is the line 2x + y = b tangent to the parabola y = ax^2 when x = 2?
a = -1/2, b = 2
never differentiate
a constant
A baseball diamond is a square with side 90 ft. A batter hits the ball and runs toward first base with a speed of 26 ft/s. (a) At what rate is his distance from second base decreasing when he is halfway to first base? (Round your answer to one decimal place.) (b) At what rate is his distance from third base increasing at the same moment? (Round your answer to one decimal place.)
a. 11.6 b. 11.6
The circumference of a sphere was measured to be 80 cm with a possible error of 0.5 cm. (a) Use differentials to estimate the maximum error in the calculated surface area. (Round your answer to the nearest integer.) What is the relative error? (Round your answer to three decimal places.)
a. 25 b. 0.012
y=x/a(bx^2+6) a. the curve has the tangent line 2y=21+19(x-3) at x=3. find A and B b. find any HA and VA
a=12, b=4 no VA or HA
abs vs local min and max
abs: can use end points these can be more than one value if they are equal a point can be abs and local
relative change
absolute change/reference value. dA/A
a(t)
acceleration. v'(t) or s''(t). =0 means velocity is constant.
Find the linearizaton L(x) off (x) =3√1 +x at x= 0 and use it to approximate 3√0.95
answer should be written as f(.05)~~ l(.05)=59/60
differential word problems
are not taken with respect to time. differentiate everything !!!
cross section of a cylinder=
area of a circle
abs. extrema exist
at critical numbers or endpoints. make sure to check critical points are in the interval
f'(c)=
avg velcoity, mean value theorm
volume is usually equal to
base area times h, trough would be w*h*1/2*L
y=(x/x-5)^1/2, find dy/dx
cant square y, only terms of x
remember
chain rule!!!! g dawg taught u this bro
d/dx sinx
cosx
other form of derative
f(x)-f(a)/x-a
f'(x)
df/dx
when asking if an absolute value is differntiable, ex: is f(x) =x^2 |x|differentiable at x= 0?
do one sided limit notations and prove that the deratives are the same at 0, don't do a separate function and solve for the derative
the radius of a sphere was measured and found to be 21 cm with a possible error in measurement of at most 0.02 cm.
dr=.02 r=21
A spotlight on the ground shines on a wall 12 m away. If a man 2 m tall walks from the spotlight toward the building at a speed of 1.7 m/s, how fast is the length of his shadow on the building decreasing when he is 4 m from the building? (Round your answer to one decimal place.).6 m/s
draw picture. remeber to make -1.7
v(t)=
ds/dt
v(t)=
ds/dt = independent/dependent
a(t)
dv/dt and d^2s/dt^2
differential
dy/dx seperate the two. dy=f'(x)dx same thing as linarization
Find the differential dy. y = tan x
dy=sec^2(x)dx dy = y and x = dx
cos(x) even or odd?
even
inflection point
f''=0 or dne, where concavity change. is a point where f is continuous and the concavity of f changes from upward to downward or downward to upward.
(f ◦ g)'
f'(g(x)) · g'(x).
evaluate limh->0 (9+h)^1/2-3/h using a derivative
f'(x)=1/2(x)^1/2 find, f('9)=1/6
If a ball is given a push so that it has an initial velocity of 3 m/s down a certain inclined plane, then the distance it has rolled after t seconds is s = 3t + 2t2.
ignore the 3. read carefully and realize the function ignores it as well
lineratization
it uses tangent line at (a, f(a)) to estimate. l(x)= f(a) +f'(a)(x-a)
"why doesnt this contradict rolles"
just say undefined, no way of knowing what
Definition of Derivative
lim h->0 f(x+h)-f(x)/h
abs. extrema of f(x)= |2-3x| -1 on (0,1) ( meant to be brackets)
max = f(0)=1 min=f(2/3)=-1
sin (x) even or odd
odd
when making function neg for abs value function
only make top or bottom negative NOT BOTH
tan period
pi
2sec^2(x)(1+tanx)=f' critical numbers
pi/2 and 3pi/2, is undefined at pi/2 also cant apply rolles of f is undefied at pi/2
when does cos(x) =sin(x)
pi/4
before taking the derative
simplify first, then take it
abs. v(t)
speed (has no direction)
g(t) = |3t − 5| critical number
t=5/3
when see absolute value
take derative from both sides to see if it is differitable. ex: sin(1/2|x|) set up in limit defintion then do the piecewise then do two different sides of limit with two different functions see if h-> exisits, if not it is DNE, if so, then it exits
d/dx
take derivative of something with respect to x. function replaces y in dy/dx. ex: y=x^2+x, so dy/dx=2x+1 is the same thing as d/dx(x^2+x)=2x+1
How many tangent lines to the curve y = x/(x + 1) pass through the point (1, 2)?
tangent line= y=m(x-1)+2 can plug in in y' for m plug in the function for y then solve for x to find number of solutions then pthing y and x coordinates
instaneous rate of change. find the instantaneous rate of change of C with respect to x when x= 100.
the deterative, when it says " respect " u can take the derative and not do limit notation
normal line
the line that is perpendicular to the tangent one
At 2:00 p.m., a car's speedometer reads 30 mi/h. At 2:10 p.m., it reads 50 mi/h. Assuming the car has beendriven the whole time, show that at some time between 2:00 and 2:10, the acceleration is 120 mi/h2.
this is the velocity function, apply mean value theorm. time shoild be put as f(0) and f(1/6)
when doing limit defintion of derivative remember
to be careful with negatives and distribute !!
derivative
top is indepenent, bottom is depenedent top is function of the bottom. takinf derative of the top with respect to the bottom
abs. s(t)
total distance travelled, u have to find critical points and plug them in as intervals. ex: Total Distance =|s(8)−s(5)|+|s(5)−s(1)|+|s(1)−s(0)|= 120 ft
y = a^6 + cos6 x
treat a as constant when finding y'
d/dx ((x+x^1/2-1)/x)
u can plug out the x to do each one separate and not use quotient rule, usually when there is one term on the bottom. bottom can also be made neg. for product. sometimes easier. remember that
when doing differentials
u should dy= (stuff) dx
remember in linarization
use equal sign thing!!
related rates constant
usually is zero bc derative is 0
problems with implicit functions is
usually messy till end
Critical numbers are
when derative is 0 or undefinded. is a value c in the domain of f such that f'(c) = 0 or f'(c) does not exist.
which the tangent line is horizontal
when f'=0
