15: Using first principles (by definition), prove thát thể đêrivätive of ä constant is U

Q: Let c>0 be a constant.  Is this result true or false?

A:

Q: Two equivalent forms of the Chain Rule for calculating the de-rivative of y = f(g(x)) are presented…

A: Given, if y=fx and u=gx then the derivative of their composition can be expressed as

Q: Find the maxamin s in flectlon Points and the in ter Vals on which the function inerease,decrease…

A: follow the solution which is mentioned in step 2.

Q: Find an upper bound for ||L|| where L: R² → R³ with L(x, y) = (2x + y,x + 3y, 3x – 2y) -

A: Here we use the definition of boundedness of a linear operator.

Q: 1. Prove that for all x, y ER, max (x, y) = x+y+lx-yl 2 x+y-lx-yl 2 and min (x, y}=-

A: What is minimum and maximum Value: For a given set of numbers, the minimum value of the set is the…

Q: 2. Let X be continous with pdf f(x) = 1/x² if 1 < x < ∞, and 0 otherwise. (a) Does E(X) exist? (b)…

A:

Q: .Prove that e- -dt < (1+ t²)* 1/4 6.

A:

Q: Verify Young's Theorem y? x2 z = In(2x + y) + +e(x+2y) y

A:

Q: [1] Determine the orthogonal trajectories of y(1+2x?)=ax.

A: The given equation is:y1+2x2=axy=ax1+2x2              ..............(1)

Q: 1.Test the function def ined by f(x.y)=xlnx- ylnx for its homogeneity. If it is homogeneous, give…

A:

Q: (3) A study on optimizing revenue from a website considered dividing cus- tomers into two groups…

A:

Q: 1+ By using the definition In(1+x) = / 1 -dt, show that for a certain range of values of r, t…

A: Solution:

Q: 9. Verify that p(t) = te/50 satisfies the condition p(t) dt = 1

A:

Q: If A(x)= f(t)dt is increasing on [0,10], then f(x) must be increasing on [0,10].

A: Given,A(x)=∫0x f(t) dt

Q: Differentiate g(N) = rN g'(N)= with respect to N. Assume that K and r are positive constants.

A: We need to find g'(N).

Q: 6) If y=x² +1+ f dt , prove that y"= 2 - x2

A:

Q: 18. Use the Linear Approximation to f(x, y) = /x/y at (9,4) to estimate 19.1/3.9.

A:

Q: If f(x,y,z) = z In(x²ycosz): then fz is

A: Here, the given function is: f(x,y,z)=z ln(x2ycos z) Using the properties of the natural logarithm:…

Q: Recall the chain rule from differential calculus: f (g (æ)) = f' (9 (æ)) · g' (æ). d dx Find de tan…

A: We have to find

Q: Consider the For at function y= In(x+1) . what valves" of x is y undefined ?

A: The given function is y=ln (x+1) We know that, for the function ln x, x&gt;0

Q: Eliminate the arbitrary constants a. x = -In t, y = at + bt?

A:

Q: Find the best linear approximation to f on th e interval [0, 1].                            f (x)…

A:

Q: 3.Find the linear approximation to f(x) = cos x at x, = ÷ and use the linear approximation to…

A: We will use formula for linear approximation.

Q: Determine k so that f(x, y) = kx(x- y) for 0<x <1,-x< y<x is a joint pdf.

A:

Q: 3) Consider a particle of mass m in a one-dimensional box of width L. Apply the variational…

A:

Q: Determine the value of the arbitrary constant of the antriderivative of F(X) = x-In(x) given the…

A:

Q: Find the best linear approximation, g, to f on the interval [0, 1]. (x) = 2/x g(x) %3D

A:

Q: Show  that the function Z =1/2(eysinx - e-ySin(x) satisfies the La Place’s equation  =0

A:

Q: Suppose that t hours after the injection of x milligrams of adrenalin the response is R units, and…

A: The response after t hours of x mg of adrenalin is R=te-t(c-x)x where c is any constant. We want…

Q: Assume that after w months, butterflies’ populace in 1000 is calculated as: f(w)=12w+cos⁡(16w)+n×100…

A: Given: Butterflies’ populace after w months is given by f(w)=12w+cos⁡(16w)+n×100 in 1000 where n=824

Q: Verify Young's Theorem S. f(x, y) = = e(2x+y) In(xy)

A:

Q: 1. Prove that for all x, y ER, max (x, y) x+y+lx-yl 2 x+y-lx-yl 2 and min (x, y)=-

A:

Q: A body of mass M1 has 1/4 times the linear momemtum of M2 if M2 has three times the mass of M1, what…

A: Momentum of the object is equal to the mass of that object multiplied by the velocity of the object.…

Q: which is the best definition of a counterexample

A:

Q: Find the linearization L(x) of the function at a. f(x) = sin(x), a = 3 L(x)

A:

Q: Verify Young's Theorem f(x,y) = e(2x+y) In(xy)

A:

Q: What is the linear approximation for any generic linear function y=mx+b?

A: Given: y=mx+b

Q: Suppose z = ƒ(x, y), where x and y are functions of t. How many dependent, intermediate, and…

A: It is observed that z=fx,y is function of x and y.

Q: Show that the linearization of f(x) = (1 + x)k at x = 0 is L(x) = 1+ kx.

A: We need a linear approximation formula for f(x).

Q: At time 0, M is deposited into Fund X, which accumulates at a force of interest δ# = 0.006t*. At…

A:

Q: . Suppose that we don't have a formula for g(x), but we know that g(2) = 4 and g'(x) = Vx² + 5 for…

A:

Q: | 28. For each x > 0, let G(x) = J each x > 0. e dt. Prove that xG(x) 1 for

A: Given information:

Q: If F(x)=[ cos(t² +1)dt , calculate using the fundamental theorem of calculus derived version.

A: Given : F(x)=∫-xxcos(t2+1)dt Aim: to calculate using the fundamental theorem of calculus ,the…

Q: Suppose it take aforceof 21 lb to stretch an industrial spring 3 ft from its equilibrium position.…

A:

Q: Show mathematically that MRS is increasing for U=1-ex +e2y and is diminishing for U=lnX +LnY

A:

Q: (b). Examine the behavior of e*+iy as x → -.

A: Euler's formula states that for any real number x  Any complex number z = x + iy,  can be written…

Q: 13. Prove that the function 2x 1 dt t F(x) is a constant on the interval (0, ∞ )

A: The student has specifically asked for the solution to Question-13. Hence, that question is solved…

Q: Find the linear approximation of f(x)=(1+x)3 at x=0. Using this approximation to estimate (1.01)3…

A: