Rates of Change
The formula for the average rate of change of a function f between x = 2 and x= 5.
(f(5)-f(2))/(5-2)
The derivative of
f(x)=3x^5-14x^3+x^e+pi^2
f'(x)=15x^4-42x^2+ex^(e-1)
The points you should check when finding the extrema of a function.
Critical points and Endpoints.
The linear approximation of f(x) centered at x=a.
L(x) = f(a)+f'(a)(x-a)
The expression that represents the total area of f(x) between [a,b]
int_a^b |f(x)| dx
The better approximation of the rate of change at x=3 between
(f(3)-f(2))/(3-2)
or
(f(3)-f(2.5))/(3-2.5)
(f(3)-f(2.5))/(3-2.5)
The derivative of
f(x) = ln(x^(1/3)+1)
f'(x)=1/(x^(1/3)+1)(1/3)x^(-2/3)
What we call a function at x=a when f(a) exists and
lim_{x \to a} f(x) = f(a)
Continuous
The linear approximation of f(x)=cos(x) centered at
x=pi/4
L(x) =cos(pi/4) +(-sin(x))(x - pi/4)
L(x) = sqrt(2)/2-sqrt(2)/2(x-pi/4)
The definite integral expression that represents
\lim_{n \to \infty} sum_{i=1}^n f(7/2 + 81/n i )(81/n)
\int_{7/2}^(169/2)f(x) dx
The average rate of change between t=12 seconds and t=8 seconds where f(12)=9 meters and f(8)=5 meters.
(f(12)-f(8))/(12-8) = 1 (meters)/sec
The partial derivative, with respect to x, of
f(x,y)=x^2cos(y)+x^y-y^x
f_x(x,y)=2xcos(y)+yx^(y-1)-y^xln(y)
The point (c,f(c)) where the output f(c) is less than or equal to every other output value on the domain of the function.
Absolute Minimum
The right Riemann Sum of f(x) from [a,b] using n subintervals.
R_n = sum_{i=1}^n f(a+i\Delta x)\Delta x
R_n = sum_{i=1}^n f(a+i(b-a)/n)(b-a)/n
The net-signed area of cos(x) over
[0,pi/2]
1
int_0^(pi/2) cos(x) dx = sin(x)|_0^(pi/2)=1
The rate of change of f(x) at
x = pi
where
f(x) = (cos(x)-3)/sin(x)
f'(x) = (-sin(x)sin(x)-cos(x)(cos(x)-3))/(sin2(x))
The derivative of
f(x) = 7/(4(5^x))
f'(x)=7/4 (ln(5)5^(-x)(-1))
The maximum value of f(x)=-x^2 -2x -1.
0.
f'(x) = -2x -2
f'(x) = 0 \Rightarrow x=-1
f(-1)=0
The left Riemann Sum of f(x) = x^2 from [0,1] using 4 subintervals.
L_4= (0.25)(f(0)+f(0.25)+f(0.5)+f(0.75))
L_4=0.421875
The Riemann Sum expression of
\int_0^31 sin(4x)dx
\lim_{n \to \infty}\sum_{i=1}^n sin(0+ 31/ni)(31/n)
\lim_{n \to \infty}\sum_{i=1}^n sin( 31/ni)(31/n)
The average rate of change of f(1,2)=18 and f(3,2)=3.
(f(3,2)-f(1,2))/(3-1)=(3-18)/(3-1)=-15/2
The derivative of
f(x)=(4/(5ln(3-x))) (3x^pi-7x+4/x
f'(x)=(4/5)(-1(ln(3-x))^(-2))(1/(3-x))(-1)(3x^pi-7x+4)
+ (4/(5ln(3-x)))(3pix^(pi-1)-7-4x^(-2))
The maximum area of a fenced in rectangular garden where the total amount of fencing is 400 feet, and one side of garden does not need a fence.
20,000 feet2
x+2y=400 \Rightarrow x=400-2y
A(x,y)=xy=(400-2y)y=400y-2y^2
A'(y)=400-4y \Rightarrow y =100 feet
The type of function where both the left and right Riemann sums equal each other.
Constant functions f(x) = c.
The amount of time a car travelled if they started at s(0) = 1 meter, ended at 26 meters, and travelled according to
v(t)= 2t
s(b)=s(a) + int_a^bv(t)dt
26=1+int_0^t 2t dt
25 = t^2 \Rightarrow t=5 s