Fundamental Theorem of Calc
What does the fundamental theorem of culculus help you find?
allows you to calulate the exact area
What is a limit?
The limit of a function describes the value a function approaches as it approaches a certain x value
What is the derivative of 2x⁴+4x
8x³+4
What is an antiderivative?
Antiderivative undoes the derivative
the function F(x) is the antiderivative to the function f(x)
What is the mean value theorem?
f_avg = (1 / (b - a)) * ∫[a, b] f(x) dx
What is the fundamental theorem of calculus (part 2)?
a∫b f(x) dx =F(b)-f(a)
When does a function have a discontinuity?
If there is a break in the graph, like a jump, undefind, infinite or removable points on the graph
What is the derivative of x²(2x+1)?
6x²+2x
what are the properties of definate integrals?
1) a∫b C dx = C(b-a)
2) a∫b C f(x) dx = C a∫b f(x) dx
3) a∫b f(x) dx = a∫c f(x) dx + c∫b f(x) dx
4) a∫b f(x) g(x) dx= a∫b f(x) dx + a∫b g(x) dx
Evaluate ∫(8x−12)(4x2−12x)^4dx
15(4x2−12x)5+c
what is the fundamental theorem of calculus (part 2)?
a∫x f(t) dt = f(x)
Find the limit of the average rate of change of f(x) from c to x , f(x)=8x³ c=2
lim x → a f ( x ) = 96
What is the quotient rule?
g(x)f´(x) -f(x)g´(x)/ [g(x)]²
what is the average value of f(x)=sin x on [0,π]
Favg= 2/π
What is the Trapezoidal rule?
Tn=△x/2[f(x0)+2f(x1)+2f(x2)..+f(xn)]
Given g(x)=-1∫x² cos t dt
Find G(x)
G(x)=2x cos x²
Given g(x)= x sin (1/x), find lim x→0 g( x ) using the squeeze theorem
Lim x→0 g( x ) = 0
What is the derivative of log 2 (x²)?
y=(1/x² ln 2) (2x)
find the average value of f(x)=4-x² on [0,3]
Favg=1
solve the differential equation dy/dx=6x²/2y+cosy
(general solution)
y²+siny=2x³+c
given g(x)= 0∫1 5/1+x² dx
find the value g´(x)
g´(x)= 5π/4
Find the equation of a tangent and normal line at (2,4), given f(x)=x²
tangent line: y-4=4(x-2) or y=4x-4
normal line: y-4=-1/4(x-2) or y=1/4x+1/2
Find y´ given y=x^x
y=x^x(1+lnx)
a particle moves along a line with v(t)=t²-t-6 m/sec find the displacement on [1,4]
total distances traveled=-4.5
Find the area between y=e^x and y=x on [0,1]
A=e-3/2