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What is the derivative of 3x?
Let y = 3x y' = 3(x)' y' = 3(x1)' y' = 3[1x(1-1)] y' = 3(1x0) y' = 3(1 x 1) y' = 3 In general: y = xn y' nx(n-1)
Matrix inverse method ------ x1 x2 x33000 -x1 5x2-x30 2x1-3x30?
inverse matrix. x1+x2+x3=3,000 -x1+5x2=0 2x1-3x3=0.
X plus one over x squared plus two?
it equals x1 it equals x1
What is the definition of a point-slope form?
point slope form is y-y1=m(x-x1). x1 and y1 are both points and m is the slope.
State and prove Fundamental theorem of integral calculus?
This part is sometimes referred to as the First Fundamental Theorem of Calculus.Let f be a continuous real-valued function defined on a closed interval [a, b]. Let F be the function defined, for all x in [a, b], byThen, F is continuous on [a, b], differentiable on the open interval (a, b), andfor all x in (a, b).ProofFor a given f(t), define the function F(x) asFor any two numbers x1 and x1 + Δx in [a, b], we haveandSubtracting the two equations givesIt can be shown that (The sum of the areas of two adjacent regions is equal to the area of both regions combined.)Manipulating this equation givesSubstituting the above into (1) results inAccording to the mean value theorem for integration, there exists a c in [x1, x1 + Δx] such thatSubstituting the above into (2) we getDividing both sides by Δx gives Notice that the expression on the left side of the equation is Newton's difference quotient for F at x1.Take the limit as Δx → 0 on both sides of the equation.The expression on the left side of the equation is the definition of the derivative of F at x1.To find the other limit, we will use the squeeze theorem. The number c is in the interval [x1, x1 + Δx], so x1 ≤ c ≤ x1 + Δx.Also, andTherefore, according to the squeeze theorem,Substituting into (3), we getThe function f is continuous at c, so the limit can be taken inside the function. Therefore, we get which completes the proof.
