Suppose you are standing in a field near a straight section of railroad tracks just as the locomotive of a train passes the point nearest to​ you, which is 1/5 mi away. The​ train, with length 1/12 mi, is traveling at 13 mi/hr. If you start running in a straight line across the​ field, how slowly can you run and still catch the​ train? In which direction should you​ run?   Let v be your velocity and let x be the difference between the position of the locomotive when it passes you and the position at which you catch the tail of the train. What is the objective function in terms of​ x?

Q: Show that an equation of the tangent line to the graph of =D2 at the point (a, b) is given by 2. b…

A: It is given that, Now take the derivative of the given function with respect to 'x'. Therefore,…

Q: Find the slope of the tangent line to the curve−4x^2+4xy−3y^3=−5at the point (1,−1).

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Q: For what value of x does the tangent line to the curve y=7+3e^x−6x  is parallel to the line 8x−8y=9?

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Q: For which value(s) of x does f(x)=−x^3/3−2x^2+3x−8 have a tangent line of slope 7?

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Q: Find a number a such that the tangent line to the graph v= 3x at x = a is parallel to the tangent…

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Q: Find (a) the equation of the secant line through the points where x has the given values, and (b)…

A: Given: f(x)=2x-1 x=5 and x=10. (a) Determine the value of f(x)=2x-1 at x=5. f(x)=25-1=24=4…

Q: Find f(4) and f'(4), assuming that the tangent line to y = f(x) at a = 4 is y = -7x +2.

A: The derivative of a function at a point represents the slopeof tangent at that point .

Q: Find the tangent line to g (x) 16 4T at z = 4.

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Q: Suppose that a particle moves along a straight line with velocity v(t) = 13 – 6t, where 0 <t < 2 (in…

A: To find the displacement at time t.

Q: A particle moves along a straight line and has acceleration given by s" = 5t. Its initial velocityis…

A:

Q: Find the slope of the tangent line to the function: 4y 27 1 + х2 At point (1, 1) . 2.

A: Given:

Q: Find (a) the equation of the secant line through the points where x has the given values, and (b)…

A: Given f(x)=3x2-5x+7, x=2, x=4

Q: Find an equation of the line tangent to f(x) at the point |3D 1+e

A: Find slope at given point and then use formula  to find tangent line

Q: If a ball is given a push so that it has an initial velocity of 24 ft/sec down a certain inclined…

A: Here we can use the fact that the velocity is the instantaneous rate of change of distance. That…

Q: Find the slope of the tangent line to the curve ((8x+9y)^(1/2))+((2xy)^(1/2))=7 at the point (2,1)

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Q: Find the equation of the tangent line to the graph of y = -2x° + 6x? – 5x + 10 at the point (1,5) in…

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Q: Find f (3) and f' (3), assuming that the tangent line to y = f (x) at a = 3 has equation y = 5x + 2.

A: Tangent to a function at a point is a line which just touches the graph of the function at that…

Q: Find the slope of the tangent line to the curve1x^2+1xy−3y^3=−24at the point (−2,2).

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Q: Find a number a so that the tangent lines to y= x° + 11x +x + 13 at x = a and at x = a+ 1 are…

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Q: Find an equation of the tangent line and normal line to the curve at the given point for the…

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Q: 8. Find the equation of the tangent line to the curve y = e* at the point (a, f (a)) when a = In 3.

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Q: Find the slope of the tangent line to the curve 3x? + xy – 4y³ = 6 at the point (1, – 1).

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Q: Find the slope of the tangent line to the curve 4x^2+1xy+3y^3=240 at the point (3,4)

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Q: Find the slope of the tangent line to the graph r=2sin(0) at the point V3, 3 11.

A: We have to find

Q: Find f(3) and f'(3), assuming that the tangent line to y = f(x) at a = 3 has equation y = 5x +2.

A: Consider the provided function, y=5x+2 Find the value of f3  and  f'3 at a=3. y=5x+2 We can write…

Q: Suppose f (x) = (x 2 − 2x) (3x + 2).What is the equation of the line normal to f(i.e., the line…

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Q: Suppose that a particle moves along a straight line with velocity v(t) = 7 – 4t, where 0 <t < 1.5…

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Q: Find the equation of the tangent line to f at x = a. f (x) = 3 − 4x,             a = 2

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Q: a) Find the equation of the tangent line to ƒ at (3, –27/17). y = -16.816x+48.198 b) Find the…

A: We find dy/dx using the quotient rule then we plug x=3 in it

Q: x4 C: y=-+ 3 x3 17 4 12

A:

Q: If the tangent line to y = f(x) at (3, 2) passes through the point (0, 1), find (3) and r(3). (3) =…

A: In this question, we need to find f(3) and f'(3).

Q: Find f(4) and f' (4), assuming that the tangent line to y = f(x) at a = 4 is y = -6x + 2. %3D

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Q: . Find the slope of the tangent line at the point P of (x² + y² )° = 4xy; P(1,1).

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Q: 2. Find the slope-intercept form of the tangent line to f(z) = Vī at 1 = 4.

A: Given,f(x)=x at x=4 We find the y-coordinate of x0=4.⇒y0=f(4)⇒y0=4⇒y0=2 Hence, P has…

Q: Suppose you are standing in a field near a straight section of railroad tracks just as the…

A:

Q: Suppose that a particle moves along a straight line with velocity v(t) = 10 - 1t, where 0 <ts 3 (in…

A:

Q: Find the slope of the line tangent to y = 3x(cos x) at (7/2, 7/2). A) -3/2 B) 3x/2 C) /2 D) -x/2

A: Consider the function  y=3x(cosx) Point : π2,π2

Q: Suppose the slope of the tangent line at any point on a curve is 242. If the curve passes thru…

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Q: Compute f '(x) and find an equation of the tangent line to the graph at x = a.

A:

Q: Find the slope of the tangent line to the curve x+y3D6 xy at the point (3,3). (a) 0 (b) -1 (c) 1 (d)…

A: To find: slope of tangent line at the point (3,3) to the equation of curve given below.

Q: Suppose that a particle moves along a straight line with velocity v(t) = 13 - 5t, where 0 ≤ t ≤ 1…

A: Here the velocity function is v(t)=13-5t   , where 0≤t≤1 (in meter per second). We have to find the…

Q: 2. Find the slope of the tangent line to the curve y? + yx² + x² – 3y? = 0 at the point (0, 3).

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Q: Find an equation of the tangent line at the point indicated. s(t) = In(7 – 3t), t= 1. %3D (t-1)…

A: Following steps help to find equation of line tangent . 1.Find the first derivative of f(x). 2)…

Q: Find the slope of the line tangent to the curve y =x-2x+ 1 at x 1 01 O 1/2 O 1/3 O 1/4

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Q: 2. Find the slope of the tangent line to the curves: (a) xy + y³ = 12 at the point (2, 2), (b) 3x4/3…

A: This question can be solved using the concept of tangent and normal using application of…

Q: Suppose that a turtle is moving along a straight line, and its displacement s(t) at any given time t…

A: Given data: The expression for the displacement is s(t)=(t-√(t2+3)+√3). The given interval is [a,…

Q: Suppose that a particle moves along a straight line with acceleration defined by a(t)=t-3 where…

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Q: Suppose that a particle moves along a straight line with velocity v(t) = 14 – 3t, where 0 <t< 2 (in…

A: Given, Velocity of the particle = v(t) = 14 - 3t  We have to find the displacement at time t = 2 So,…

Q: The slope of the tangent line to the graph of xy = In(x² + y² ) at the point (1,0) is :Select one…

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Q: Find the slope of the tangent line to the curve 1x^2+1xy−2y^3=160 at the point (−4,−4)

A: The objective of the question is determine the slope of the tangent linen to the given curve.

Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differential equations.