Plot c(t) = (sin 2t, 2 cos t) for 0 ≤ t ≤ π. Express the length of the curve as a definite integral, and approximate it using a computer algebra system.

Q: Find the points on the cardioid r = 1−cosθ at which there is a horizontal tangent line, a vertical…

A: Consider the given: r=1−cosθWrite the cartesian…

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Q: Find the length of the curve using formula.  y= ln (cosx) bounded by 0 and pi/4

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Q: Evaluate the definite integral. sin(3t)dt

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Q: Find the equation of the tangent line to the graph of f(x) = 9 sin(x) + 4 cos(x) at x = in in the…

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Q: 2 ind the arc length for f(x) = 6-2 x² over the interval [0, 2].

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Q: Find the MacLaurin polynomial of degree 7 for the function sin (ot) dt. F(X) = %3D

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Q: Evaluate the definite integral. sin(2t)dt

A: We will solve the following.

Q: Evaluate the definite integral. sin x dx -dx J12 I+cos² x

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Q: Use all your background to find the integral of the hyperbolic function: 00 sinh ax cosh bx

A: Given: The given integral ∫0∞xnsinhaxcoshbxdx

Q: The slope of the tangent line to the curve r=cos20 at 0= 1/4is none of these O-V3 0 1 O (-1)/V3

A: As we know that; The slope of the tangent line to the polar curve at the given point is equal to the…

Q: Evaluate the definite integral: r1 arctan x dx = x2 +1 0.

A: To solve the integral first I have make a substitution so that integral become easy and the solved…

Q: 2. F(x) = [ cosr' dt

A: Use fundamental theorem of calculus

Q: Find the line tangent to the graph of f(x) = cos(x) at xo = T.

A: f(x)= cos(x)  Then, f(π) = cosπ⇒f(π) =-1

Q: ÿ + ỷ = sec t tan t

A: This is the question of the differential equation.

Q: 3 sin (t) cos“ (t)dt

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Q: Calculate the derivative e10x de L9 In (t)dt using Part 2 of the Fundamental Theorem of Calculus.…

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Q: nd the points at which f(x) = cos² x has a horizontal tangent line. %3D

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Q: Use the Fundamental Theorem to evaluate the definite integral exactly. 16 Enter the exact answer. 16…

A: Given query is to find the value of the integral.

Q: Locate the first nontrivial root of sin(x) = x² where x is in radians Use bisection with the initial…

A: Given sinx=x2

Q: Find an equation of the tangent line to the graph of f(x) = | t cosh (t² – 1) dt tan-lt 1+In (x) at…

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Q: Use the Newton-Raphson Method to approximate the root of f(x) = 2 cos(x) – 2x + 1.

A: Newton Raphson method: numarical methods

Q: Find the points on the cardioid r = 4(1 – sin0) at which there is a horizontal tangent line, a…

A: According to the given information, it is required to determine the horizontal tangent line,…

Q: the equation of the tangent line to the graph of f(x) = 9 sin(x) + 4 cos(x) at x = in the form y =…

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Q: The slope of the tangent line to the curve parametrized by the equations x (t) = -2(cos(t) + t…

A: Slope of tangent is equal to dy/dx.

Q: a(22) V17 – cos(2nt)" (2 – 4) sin() 1. Given f(x) dt, find the equation of the line tangent to the…

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Q: Calculate the length of the curve r=a*sin(^3)(x/3) for (x(1)=0, x(2)=3pi)

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Q: Find the arc length of the graph of the function over the interval. y = In(sin z);

A: Given,          y=lnsinx ; π4,3π4

Q: 4. Use definition (not tables) of L {f(t)}= [ e"f(t)dt to find L{sin(2t)} . %3D

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Q: 1. Determine the equation of the tangent line to the graph of f(a) = [² pr+sin r = x = 0. cos (t² +…

A: Above equation of tangent is given.

Q: Find the exact length of the curve. y = 2 In(cos (÷)) Tt/3 < x <T/2

A: Here given Differentiate the given function with respect to x using the chain rule of derivative

Q: Find the exact length of the cur curve. y = Vx- x2 + - sin (V)

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Q: Use the Intermediate Theorem show that there is a root of the given equation: sin a - x between…

A: Intermediate value Theorem states that if f is continuous on the interval (a ,b ) and L is a number…

Q: Find the length of the catenary y = a cosh from x = 0 and x = a X1.

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Q: Determine the equation of the tangent line to the graph of g(x) = Sm, [1 – sin(ne')]dt at %3D In x…

A: We have to find the equation of the tangent line to the graph of gx=∫lnx01-sinπetdt at the point…

Q: Evaluate the integral.  integral sin 3  x cos 2 x dx

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Q: Determine the length of the curve TC f(x) = In[sec(x)],0 < x < 4

A: The given curve is- f(x)=lnsec(x)Interval 0≤x≤π4We need to determine the length of the given…

Q: Evaluate the definite integral.

A: ∫012arccosx1-x2 dx Consider, ∫arccosx1-x2 dx Now put, arccosx=t⇒-11-x2=dtdx⇒11-x2 dx=-dt Therefore,…

Q: Find the lengthof the curve r= a sin (0/2), 0s 0S T, a>0

A: The formula for finding arc length is L = ∫0πr2+ drdθ2dθ, where r = a sin2(θ2)

Q: Evaluate the integral. 2 sin -1 x dx

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Q: 3) Find the exact length of the curve. y = In(cos r), 0 < x<

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Q: Consider the function f(x) = cos x − 3x + 1. Since f (0)ƒ () <0, f(x) has a root in [0]. If we use…

A: Here f(x) is a continuous function . If f(a)f(b)&lt;0 then f(x) has a root in [a,b]. Now we use…

Q: Find an (1,1) equation of the tangent i ine at the curue yt+xy = x_x+a. on

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Q: 4. Use definition (not tables) of L{f(t)}= [ e"f(t)dt to find L{sin(2t)} .

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Q: Q1. Find the length of the of the curve y = In(cos x) from x = 0 to x = . %3D

A: Explanation of the solution is given below...

Q: arc sin z = -i log(iz + V1- 22).

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Q: (a) Evaluate the integral t tanx dx by using the reduction formula of tangent function.

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