2. Two stations, located at M(-1.5, 0) and N(1.5, 0) (units are in km), simultaneously send sound signals to a ship, with the signal traveling at the speed of 0.33 km/s. If the signal from N was received by the ship four seconds before the signal it received from M, find the equation of the curve containing the possible location of the ship. Graph the equation in a clean sheet of paper.

Q: 1 ) Find the equation of the curve for which   y” =  6x – 4 and which passes through (1,2) and (-2…

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Q: Find the equation for the curve in the xy-plane that passes through the point (1, -1) if its slope…

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Q: Find the shortest distance from the point x = (-3, 0, -12)  to the line r = t(0, -2, -2).

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Q: Find the equation of a line normal to the graph of y = 3x+5/x−1 at x = 3

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Q: Suppose a curve is defined by the equation 16(x² + y²) = 45. Show that if you translate the curve to…

A: we have to translate the curve.

Q: Hello. Can you please explain this problem to me?

A: Slope of the function is the derivative of the function.

Q: 2.1 Identify and sketch the graph of y = /9 – 4x². Show all the necessary points on the curve and…

A: We can square the left-hand side and right-hand side then we get y2 = 9-4x2

Q: Suppose a curve is defined by the equation 16(?2+?2)=45.16(x2+y2)=45. Show that if you translate the…

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Q: Find an equation of the curve that passes through the point (1, 3), and has the given slope y′ =…

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Q: Find the exact length of the curve = √(y - 3), 1 ≤ y ≤

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Q: Find a parabola with equation y = ax2 + bx + c that has slope 8 at x = 1, slope -12 at x = -1, and…

A: To find a parabola with equation.

Q: Find an equation of the curve that passes through the point (0, 2) and whose slope is xy?

A:

Q: Find an equation for the line perpendicular to the tangent to the curve y = x°- 16x + 6 at the point…

A: Given: y=x3-16x+6, 4,6 we know that equation of line which perpendicular to tangent at given point…

Q: the slope of the curve 2y2=ax2+b at (1,-1) is -1.find a,b

A: The equation of the curve is  2y2 = ax2+b…

Q: The equation of the line tangent to the curve y=3x+4/4x-3 at the point (1,7) is  a) y - 25x=-18 b) y…

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Q: Graph the relation of equation x2/4 + y2/16 = 1 using the points calculated in the table below.…

A: Answer

Q: Find the slope of the curve 1 at the point P (2, ;).

A: Explanation of the answer is as follows

Q: Find a parabola with equation y = ax² + bx + c that has slope 4 at x = 1, slope -16 at x = -1, and…

A: We know slope of a curve y=f(x) is given by, dydx=f'(x)

Q: What are the coordinates of the points in the graph of y-x+6x-13=0 in which the tangent line is…

A: Given query is to find the point where tangent line is horizontal.

Q: Find the length of the curve with equation x2 (y + 1)3 from the point where æ = 0 to the point where…

A: Since the given curve is  x2=y+13      from point x=0 to x=33. Given curve can be written as x=y+132…

Q: Find an equation of the curve that passes through the point (3, 1), and has the given slope y′ = −…

A: Given that y'=-y5xdydx=-y5x

Q: Find the length of the curve defined by y = 3z3/2 - 3 from x = 3 to a = 9. The length is

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Q: Find the equation of the lines tangent to the curve x2 + y2 = 4 and parallel to the y-axis ?

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Q: 6. Find the member of the curve whose slope at any point is √y and passes through the point (2, 1)

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Q: Find the exact length of the curve y3 = 22 from (-1,1) to (8, 4).

A: Use formula of length of curve

Q: What are the coordinates of the points in the graph of y2-x²+ 6x – 13=0 in which the tangent line is…

A:

Q: Find the equation of the curve that passes through the point (3, 2) if its slope is given by y = dy…

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Q: For the curve y = ax³ + bx² + 12x - 7, what values of a and b will make the tangents to the curve at…

A: We have to solve the problem

Q: 4)Solve for the family of curves of straight lines with y-intercepts thrice as their corresponding…

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Q: 4. Find the member of the curve whose slope at any point is and passes through the point (3, 4)

A: Given slope (m) = 3x/4y Passing through a point (3,4) Therefore member of the curve whose slope…

Q: Find the member of the curve whose slope at any point is 2e** and passes through the point (0, 1)

A: We need to find the member of the curve whose slope at any point is 2e^(4x) and passes through…

Q: Find the equation of the curve that passes through the point (8, 6) if its slope is given by dy =6z-…

A: We have to find y

Q: Graph the relation of equation x2/4 + y2/16 = 1 using the points calculated in the table below.…

A: We have to graph the function x24+y216=1 by using the points in the table.

Q: Find the exact length of the curve. + 4х 1 < x < 3 y

A:

Q: Find an equation of the curve that passes through the point (1, 1), and has the given slope y′ = −…

A: y'=-9x16yPasses through (1,1)Formula∫xndx=xn+1n+1+C

Q: End the Vellue of g) if the slope of the Curve y= x-2x is (2)?

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Q: The curve is given by 2 + 1), 0<x<3. 3

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Q: What is the equation of the line tangent to the graph of y = (2x+3)/(3x-2) at the point (1,5)? 13x -…

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Q: Find a parabola with equation y = ax2 + bx + c that has slope 5 at x = 1, slope −11 at x = −1, and…

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Q: Find the length of the curve defined by y = 4x3/2 + 11 from x = 2 to x = 8.

A: We will differentiate the function and use the arc length formula.

Q: The points at which the graph of the equation 8x2+ y² – 16x + 8y + 8=0 has horizontal tangent lines…

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Q: Find an equation of the curve that passes through the point (1, −2), and has the given slope y′ =…

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Q: I Discuss the properties of the curve and sketch fully: a) y = x² e-x

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Q: Determine the length of the curve x = (y-1), where 4 <y <9.

A: Now differentiating given curve with respect to y, we get

Q: Find the length of the curve for 12x = 4y³ + 3y-¹ where 1 ≤ y ≤ 3 Enter your answer in exact form.…

A:

Q: 4. Find the equation of a curve whose slope at any point is equal to x + + and which passes through…

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Q: Use the methods of this section to sketch the curvey =  x3  -3a2x +2a3, where is a positive…

A: The curve is  y =  x3  -3a2x +2a3, where is a positive constant.

Q: What are the coordinates of the points in the graph of y-x²+6x – 13=0 in which the tangent line is…

A: Since you have asked multiple questions in a single request, we would be answering only the first…

Q: Find the slope of the curve x3y3 + y2 = x + y at the points (1, 1) and (1, -1).

A: Given,

Q: or the curve y = ax³ + bx² + 12x - 7, what values of a and b will make the tangents to the curve at

A:

Arc Length

Arc length can be thought of as the distance you would travel if you walked along the path of a curve. Arc length is used in a wide range of real applications. We might be interested in knowing how far a rocket travels if it is launched along a parabolic path. Alternatively, if a curve on a map represents a road, we might want to know how far we need to drive to get to our destination. The distance between two points along a curve is known as arc length. 

Line Integral

A line integral is one of the important topics that are discussed in the calculus syllabus. When we have a function that we want to integrate, and we evaluate the function alongside a curve, we define it as a line integral. Evaluation of a function along a curve is very important in mathematics. Usually, by a line integral, we compute the area of the function along the curve. This integral is also known as curvilinear, curve, or path integral in short. If line integrals are to be calculated in the complex plane, then the term contour integral can be used as well.     

Triple Integral

Examples: