An object moves in a straight line so that its velocity after t seconds is v(t) = 2/t – t. Determine the displacement and distance traveled by the object on [0, 16].

Q: 4. Find the equation for the line tangent to a. y=x²- 3 x + 5 at the point (2, 3)

A: We will find tangent equation.

Q: Find an equation of the secant line containing (1, f(1)) and (2, f(2)). f(x) = x² - 2x %3D

A: Given , the function f(x) = x2 - 2x Now we find      f(1) = 12 - 2•1 = 1 - 2 = -1,…

Q: calculate the length of the curve y=2cosh (x/2) for 0 < x < 2

A:

Q: Find the point(s) on x=3-y² that are closest the point A=(1,0)

A: Our Aim is to find the points on x=3-y2 that are closest the point A=(1,0).

Q: Two objects moving along x-axis are starting at the same time. Their positions are measured in…

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Q: Find equations of both lines through the point (2, –3) that are tangent to the parabola y = x2 + x.

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Q: Find the slope of the line tangent to the graph of x^2+2xy^2+3y=31 at the point(2,-3)

A: The slope of the tangent line can be calculated by using the derivative of the function at the point…

Q: Find an equation of the secant line containing (1, f(1)) and (2, f(2)). f(x) = x² - 2x

A: Rule:-equation of line passing thraught (x1,y1) and (x2,y2) is y-y1=y2-y1x2-x1(x-x1)

Q: An insect is moving on the surface of a hot plate, the temperature of which at the point x units to…

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Q: A coordinate of the point on the curve of y=2x3−3x−6, where the tangent at the point is…

A:

Q: An insect is moving on the surface of a hot plate, the temperature of which at the point x units to…

A:

Q: Find the equation of the line segment tangent to the curve y−2x2−1=0y−2x2−1=0 which is parallel to…

A:

Q: A particle is moving along a horizontal line according to the equation s = 3t2 − t3, t ≥ 0, where s…

A: Instantaneous velocity is given by the derivative of position function. The particle is moving to…

Q: Determine the coefficients a, b, c and d so that the curve y = ax³ + bx² + cx + d be tangent to the…

A: Curve y=ax3+bx2+cx+d is tangent to the line y=3x+8 at the point (-2,3). The curve y=ax3+bx2+cx+d has…

Q: Find the point on the line y = 3x + 4 that is closest to the origin. (x, y) = |

A: To find - Point on the given line closest to the origin.

Q: Find the equation of the line segment tangent to the curve y−2x^2−1=0 which is parallel to the line…

A: Given curve is y-2x2-1 = 0

Q: (b) Find the equation of the tangent line to the curve 8x²y3 – 5x? + 7y³ = 10(x – 3y), at the point…

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Q: Find the tangent plane to the equation 4x2 + 3y2 – y at the point (-4,-1,68) Z =

A:

Q: Find the point on the line y=5x+3 that is closest to the origin (x,y)=(______)

A: Let (m,n) be the point on the line which is closest to the origin.Put n = m, y = n, in y = 5x + 3…

Q: Find the equation of the line tangent to 3x²y + 4xy2 = 6x² + 5y at the point (1,).

A:

Q: Find the points on the curve y = 2r3 – 3x2 – 12x + 20 - where the tangent is parallel to the x-axis.

A: A straight line parallel to x-axis have slope 0.

Q: A particle is moving along a line in such a way that v centimeters per second is the velocity of the…

A: A particle is moving along a line in such a way that v centimeters per second is the velocity of the…

Q: (b) Find the points on the curve y = 2x3 – 3x2 – 12x + 20 where the tangent is parallel to the…

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Q: The fictional city of Torontino radially has a population density of 1000e-0.017² people per km²,…

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Q: Find equations of both lines through the point (2, −3) that are tangent to the parabola y = x2 + x.

A:

Q: Find the point on the line y = 2x + 3 that is closest to the origin. (x, y) =

A: The perpendicular drawn from the origin to the line represents the shortest distance.

Q: Find the equation of the tangents: y=x² + 6x² + 10x + 8 parallel x-y= 2. to the line А.…

A:

Q: Find the point on the line y = 4x +5 that is closest to the origin. (x,y) = ____

A: Find the point on the line y = 4x +5 that is closest to the origin. (x,y) = ____

Q: Find the slope and equation of the line tangent to f (x) = x2 +9 at the point (1, 10).

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Q: With center on the line y = 2x – 13 and tangent to the line y = - 3x + 57 at (16, 9)

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Q: Find the slope of the curve at the point (2,3) * x³ 2x + 1 3/2 1 O 1/2 O 1/4

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Q: Find the length of the curve 9y2=8x3 from x=o to x=12 when y>0

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Q: Find an equation for the linear function g(z) which is perpendicular to the line 7z – 6y = 12 and…

A: Given the equation of a line 7x−6y=12

Q: Find the point on the line y = 3x + 4 that is closest to the origin. (x,y)=

A:

Q: An insect is moving on the surface of a hot plate, the temperature of which at the point x units to…

A: Definition used-   To keep the temperature constant, the insect has to move in a direction that is…

Q: Find the equation of the curve, in the XY plane, that passes through the point (2, 2) and is such…

A: Differential equation

Q: An insect is moving on the surface of a hot plate, the temperature of which at the point x units to…

A: As a first step we need to find the gradient of the surface and evaluate the same for the point (1,…

Q: 4) Find an equation to the line tangent to y = 5 + |x – 2| at the coordinate (2, 5).

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Q: Find the point(s) on the curve of y = 3x^4 − 4x^3 where the tangent line is parallel to the line…

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Q: Find the length of the curve on the interval 3/2 Y= (10x+ 15) 15

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Q: Find the slope for f(x,y) = xeyat P (2,0) in the direction of the line from M(2,0) to N(-1,2)

A: We have to find slope

Q: Find the linear velocity of a point moving with uniform circular motion, if the point covers a…

A: From the data: s=45 cm t=5 sec

Q: Find the curve y = ƒ(x) in the xy-plane that passes through the point (9, 4) and whose slope at each…

A: Given the curve y = ƒ(x) in the xy-plane that passes through the point (9, 4) and whose slope at…

Q: Determine the coefficients a, b, c and d so that the curve y = ax³ + bx² + cx + d be tangent to the…

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Q: ]Find the x-intercept of the curve that passes through the point (0,2) and whose slope at (x,y) is…

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Q: Find the equation of the line tangent to 8x2 + 18y3 = 152 at x = 1. %3D

A: The equation of a line tangent to a curve, y = f(x) passing through the point (x1,y1) is given by: y…

Q: find an equation of the tanfent line to the curve at the given point y = 4x – 3x², (2, –4)

A:

Q: Find an equation of the line tangent to the curve y = v4x – 3 – 1 and perpendicular to the line x +…

A:

Minimization

In mathematics, traditional optimization problems are typically expressed in terms of minimization. When we talk about minimizing or maximizing a function, we refer to the maximum and minimum possible values of that function. This can be expressed in terms of global or local range. The definition of minimization in the thesaurus is the process of reducing something to a small amount, value, or position. Minimization (noun) is an instance of belittling or disparagement.

Maxima and Minima

The extreme points of a function are the maximum and the minimum points of the function. A maximum is attained when the function takes the maximum value and a minimum is attained when the function takes the minimum value.

Derivatives

A derivative means a change. Geometrically it can be represented as a line with some steepness. Imagine climbing a mountain which is very steep and 500 meters high. Is it easier to climb? Definitely not! Suppose walking on the road for 500 meters. Which one would be easier? Walking on the road would be much easier than climbing a mountain.  

Concavity

In calculus, concavity is a descriptor of mathematics that tells about the shape of the graph. It is the parameter that helps to estimate the maximum and minimum value of any of the functions and the concave nature using the graphical method. We use the first derivative test and second derivative test to understand the concave behavior of the function.