(144 – t)a" + 5x' + 7x = 0, ¤(0) = 1, x'(0) = 0. ||

Q: y(x) = tan{/3x² + In(5x*) - In(5x'

A: Given  y(x) = tan3x23 + ln5x4 We have to find dydx.

Q: The equation of tangent line of the function 21 x3 f(x) = e7e dt %3D 21 x at x=1 is V21 e3087 (x-1)…

A: Given function is  f(x)=∫21x21x3e7t2dt Since at x=1, upper limit of the integral is equal to lower…

Q: Given f(x) = 8x3 + In(x – 1). Estimate the value of f(2.01) using differential. %3D

A:

Q: Q:: If In /x2 + y? =sec(), use implicit differential to proof the following: - -x(²+y²). sec̟. tan?…

A:

Q: 7. Find the dy/dx of * y = In sin 5x

A:

Q: 3. (a) Compute ,12x – 1| dx in two ways. i. By using properties of the definite integral and the…

A:

Q: I = | In(ax + 1) dx, for a > 0

A: Find the integration of I

Q: 4) a²y=+52y =14 yat (2,1) Find equátion of tangent?

A: Equation of tangent  y-y1=m(x-x1) (x1,y1)are the points at which equation of tangent is to be…

Q: 7) Find the instantaneous rate of change of f(x) = ecos(z") + sin(In(8²)) at x = 1.

A:

Q: Prove that sinh−1 t = ln(t + √t2 + 1 ).

A:

Q: 10. Use differentials to estimate to estimate sin(46°). vZ(n+180) А. D. VZ(T+180) 360 180 В.…

A: Let y = sin (x)

Q: d Prove In Jax| dx In x for any constant a. dx

A:

Q: Drove that 3 e dx converges to 9 3e

A: Explanation and solutions is given below.....

Q: 1/2 2. Approximate the integral: 1 - cos(2x) dx within an error of 0.01. x2

A: According to question given that∫0121-cos2xx2dx within an error of 0.01

Q: Q,7 Evaluate the integral 1) L,x* dx 2) xVx+1 dx 3) f cos2xdx

A: we have to solve

Q: 3. Find a result by using homogenous for (x³ - 3x²y)dx- (x³y³)dy = 0

A: According to guidelines I have to provide answer only for  first qiestion...Thanks

Q: 5. x- + Px = sin t %3D dt

A:

Q: At time t = 0, then th

A: P(t)=0.5t4-16t3+144t2       , 0≤t≤12At t=0 the sale begins,costco is having a 12 hour sale,The rate…

Q: /4 - x2 the integral | dx, becomes

A: Given the integrationI=∫4−x2x2dx

Q: 4 sin (2x) dx cos (2x)t4

A:

Q: Find the general solution using Reduction of Order: y''+y'=cos 3x

A:

Q: From the differential equati on by eliminating the arbitrary function ffrom xyz = f (x?+y'+z°).

A:

Q: A particle moves along the x-axis with velocity given by v(t) = 6T sin (2nt) for time t > 0. If the…

A: Given: vt=6πsin2πt at t = 1, x = -4

Q: 8. y = In(2e-t sin t)

A:

Q: Q2: - If y = 3sec3t and x = etan3t prove that: - dy (In3)² dx y logy x * * -

A:

Q: 5. Find the derivatives (a-b) and the limit (c): (2) ± √( 1 cos (1²) dt) (0) ² (154) dt dx 1 r2+x…

A:

Q: -2.1 1- e* Use Simpson's rule to approximate -dx. x² + 2x – 1 1.2

A: This is a problem of numerical Integration.

Q: -TL 4.) x² sin(xy) dydx

A: Given ∫0π∫01x2sinxydydx

Q: dx J x2-4x+5

A: Since you have asked multiple questions, we will solve the first question for you. If you want any…

Q: -Tsin(nx)cosh(3y) dx + 3cos(7x)sinh(3y) dy =0

A:

Q: 53. df = (x+ 1)ye"dx + xe*dy

A:

Q: Let e 2* = 2 find dy dx at x = 1

A:

Q: Evaluate the second derivative of mq4.5.png os (bVx³) f(x) = a ln sin(cx2) if a = 21, b = 0.5, and c…

A: Given that, f(x)=alncosbx3sincx2a=21, b=0.5, c=5Substituting the values of a, b and c, we…

Q: Find xe dx x²

A: let  I= ∫0∞xe-x2 dxput   t= x2    ⇒   dt=2x dx      as x=0   ⇒ t=0 and     x=∞     ⇒ t=∞     I= ∫0∞…

Q: y" +y' = cos 3x

A:

Q: In (6x) to x if y = sin (5e') dt.

A:

Q: 22. Find OY if y2 - 4xy + x² = 10. dx

A:

Q: 4- 13 + 6y = 0 dx

A: We have to be find

Q: Eliminate the arbitrary constants x = -ln t , y = at + bt^2

A: given  x=-ln t , y=at+bt2 claim- to eliminate the arbitrary constants

Q: Q3) Find dy/dx if 1- y= In /x 2- y3xsin

A: 1) d(ln(x))dxApply the chain rule: 12ln(x)d(ln(x))dx

Q: 18. Show that Vxª + x and Vx* x→o by showing that they both grow at the same rate as x² as x' grow…

A:

Q: dy of y = sin(3x² + 5x – 4) * In(e*) is dx

A:

Q: se N.P.M. to estimate the root of -X = e -X for an initial of Xo =0

A: Here f(x)=e-x-x,  x0=0. We estimate the root of f(x)=0 by using N.R.M.

Q: y²

A:

Q: how can i derive this equation in 3 D? *be aqy ay az at Əx

A: Taking heat in three dimensions (I) Net amount of heat conducted into dxdydz per unit time.…

Q: Find an equation of the tangent line to the graph of {(=) = [. t cosh (t² – 1) dt f(x) /1+ln (x)…

A: First I have written the rule of derivative of an integral to find the slope of tangent line for…

Q: Determine the equation of the tangent line to the x³ + ln x graph of f(x) = sil 3 3 + 5tº dt at x =…

A:

Q: Show that d/dx(ln kx) = d/dx(ln x), where x > 0 and k is a positivereal number

A:

Family of Curves

A family of curves is a group of curves that are each described by a parametrization in which one or more variables are parameters. In general, the parameters have more complexity on the assembly of the curve than an ordinary linear transformation. These families appear commonly in the solution of differential equations. When a constant of integration is added, it is normally modified algebraically until it no longer replicates a plain linear transformation. The order of a differential equation depends on how many uncertain variables appear in the corresponding curve. The order of the differential equation acquired is two if two unknown variables exist in an equation belonging to this family.

XZ Plane

In order to understand XZ plane, it's helpful to understand two-dimensional and three-dimensional spaces. To plot a point on a plane, two numbers are needed, and these two numbers in the plane can be represented as an ordered pair (a,b) where a and b are real numbers and a is the horizontal coordinate and b is the vertical coordinate. This type of plane is called two-dimensional and it contains two perpendicular axes, the horizontal axis, and the vertical axis.  

Euclidean Geometry

Geometry is the branch of mathematics that deals with flat surfaces like lines, angles, points, two-dimensional figures, etc. In Euclidean geometry, one studies the geometrical shapes that rely on different theorems and axioms. This (pure mathematics) geometry was introduced by the Greek mathematician Euclid, and that is why it is called Euclidean geometry. Euclid explained this in his book named 'elements'. Euclid's method in Euclidean geometry involves handling a small group of innately captivate axioms and incorporating many of these other propositions. The elements written by Euclid are the fundamentals for the study of geometry from a modern mathematical perspective. Elements comprise Euclidean theories, postulates, axioms, construction, and mathematical proofs of propositions.

Lines and Angles

In a two-dimensional plane, a line is simply a figure that joins two points. Usually, lines are used for presenting objects that are straight in shape and have minimal depth or width.