Finding r from r' Find the function r that satisfies the given condition. r'(t) = ⟨e t, sin t, sec2 t⟩; r(0) = ⟨2, 2, 2⟩

Q: 5. S sec(3x) tan(3x) dx

A: formula:∫ secx tanx dx =sec x +Cwe are given;I=∫ sec(3x) tan(3x) dx3x =t3 dx =dtdx =dt3

Q: integration by parts to establish the reductionformula. ∫xn cos x dx = xn sin x - n∫ xn-1 sin x dx

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Q: y =V V secxtanx; y 2

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Q: sine a, +e-rag – 2cos@sinpas v x F(1,"/2, "/3 =?

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Q: Separation of Variable 1. (cos' x-sin x+ sin' x dx+sin x sin y(1- sin x \dy = 0

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Q: EXERCISE 2 DIFFERENTIATION OF FUNCTIONS Direction: Solve by showing your solutions: I. Differentiate…

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

Q: QI. Find the length of the of the curve from x = 0 tox = 3.

A: First differentiate  wrt xand then use formula

Q: 8. Gradient orthogonal to tangent Suppose that a differentiable function f(x, y) has the constant…

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Q: Using the differentiation with the first principle or the long method of differentiation, find for…

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Q: Separation of Variable (cos'x-sin x+ sin' xkx + sin x sin y(1- sin x)dy = 0

A: Separation of variables

Q: rection where the directional derivative is minimum for the function f(x,y) = xy+x²- (2,1).

A: The given question is taken from the vector calculus in which we have to find the directional…

Q: implicit differentiation to find y from dy dx 3. use 3•X 3..x• siny - Sy = = le

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Q: Implicit differentiation Find dy/dx when F(x, y) = sin xy + πy2 - x = 0.

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Q: sec (In x) + sec2(In x) tan(In x) 2х Given: dx Find: du tan(in x) 2х sec³(In x) + sec²(In x) sec(in…

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Q: DERIVATIVE OF A TRANS FORM 2t Lte sen6t

A: Use Laplace Transform table:   Ltkft=-1kdkdskLft

Q: The curve y = tan x crosses the line y = 11x at a non-zero x-value between x = 0 and x = 2 Use…

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Q: sin x 1. Differentiate the function: Y =e y = e* Inx

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Q: Tangent line given Determine the constants b and c such that the line tangent to f(x) = x² + bx + c…

A: Here we have to determine the constant a and b. The given function is f(x)=x2+bx+c…

Q: y=sin-1(x/x^2-1)x=3^1/2 find the equation of the tangent line

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Q: Arc length Find the length of the curve y = In x on the interval [1, e*).

A: Given: The curve y=lnx,  for x∈1, e2.

Q: Tangent line given Determine the constants b and c such that the line tangent to f(x) = x 2 + bx + c…

A: Given: Equation of curve: f(x) = x2 + bx + c . Equation of tangent to this curve at x=1 is y=4x+2.

Q: Formula, Arc Length = | /1+ [f'(x)]² dx. Where f'(x) denotes the first derivative of f(x). Given the…

A: The formula for the arc length of a function f(x) bounded by the interval [a,b] is: Arc…

Q: Average velocity Find the average velocity of a projectile whose velocity over the interval 0 s ts i…

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Q: Using Integration By Parts,x: x dx = u v– Jv du X sec Svou -tanx - Jsec Find Ex tan 5 2 1. X sec x…

A: The given Integral is: ∫x sec 12x tan 12xdx

Q: Equation of line pass through the point (1,1,1) and parallel to z- axis is: x=0 ,y=t , z=t+1 O x=1 ,…

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Q: f(x) = 2x sin(x), [0, 2x] (a) Find the first and second derivatives of the function. f'(x) = f"(x) =…

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Q: slope of the line tangent to the curve cos(5x-4y)-x(e^-x)=(-7pi/2)*(e^(-7pi/2))at the point…

A: To find the slope we need dy/dx We differentiate both sides of the given function with respect to x.…

Q: f(x) .3 II

A: Given that function,     f(x)=5x3   f(x)=5x-3

Q: QI) Find and For F(x.y) = where x = y = .using a) Direct For F(x,y) = ar method by substitution x…

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Q: sec7(t) tan(t) dt Evaluate indefinite anti-differen

A: We have to find the integral

Q: Consider the function f(x)=-/2·cos(x)-x+÷ on the open interval (-x , T). Find the intervals of…

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Q: sin -,x 0 Q: Let f(x) =- 0 ,x = 0 Find the set of values of m for which (i)f(x) is continuous at x =…

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Q: sin 2x+sin x dx Vsin2x -cos x v.

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Q: differentiation to find in terms of x and y. dx In(y) + 3xy - 4 = x² - 2y 5. Use implicit

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Q: Arc length Find the length of the curve 3 x² + y V3 -sin from x = 0 to x = 1. V3

A: y=x23-x2+32sin-1x3 Differentiating with respect to x, we get…

Q: tanx - tany = sin(x+y)/cosxcosy

A: The given identity to be proved is,

Q: sin x In x Differentiate the function: Y =e

A: The given function is y=esinxln x. Product Rule: ddx(f(x)·g(x))=f'(x)·g(x)+g'(x)·f(x) Chain Rule:…

Q: *Implicit Differentiation Directions. Use implicit differentiation to find dy dx 1. x2-y2 8 %3D

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Q: derivative of f(x, y, z) = x + z at P(6,5,-5) in the direction from P to Q(5,7, -4). NOTE: Enter the…

A: We can solve this as follows:

Q: y = Vsin (x)+ /sin(x)+. sin (x)+ sin (x)+ Vsin (x)+

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Q: Equation given : x-(z^2)=arctan(yz) Find ∂z/∂x by using implicit partial differentiation

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Q: FIRST DERIVATIVE: Four-point central y = sin(0.5 Vx)/x at x = 1, h = 0.2 Your answer

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Q: Use a computer algebra system to analyze the function over the given interval. 1 f(x) = sin(x) –…

A: To find the derivatives of given function

Q: Find the rate of change of the function fin the direction of PA. Also find the maximum value of that…

A: Given: fx,y,z=x+z2+3yexz-zcosxy, A2,1,0, P(3,2,1)

Q: Let f(x, y) = x² + xsin(y). %3D

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Q: sec (In x) + sec2(In x) tan(In x) dx Given: 2x Find: du

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Q: sin 3x .cot 5x lim x→0 X cot 4x 2 t+ sin t lim t + co s t

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Q: Consider the function on the interval (0, 27). sin(x) f(x) 4 + (cos(x))² (a) Find the open intervals…

A: Find out the intervals on which the function is increasing or decreasing and also find out the…

Q: The curve y = tan x crosses the line y = 11x at a non-zero x-value between x= 0 and x = 2 Use…

A: Newtons method of approximation is  xn+1=xn-f(xn)f'(xn) y=tanxy=11x we need to find the x value…

Q: Average velocity Find the average velocity of a projectile whose velocity over the interval 0 ≤ t ≤…

A: Given, Velocity function, v(t)=10sin(3t), 0≤t≤π