Given the function f 1x2 = 1 - cos x and the points A1p>2, f 1p>222, B1p>2 + 0.05, f 1p>2 + 0.0522, C1p>2 + 0.5, f 1p>2 + 0.522, and D1p, f 1p22 (see figure), find the slopes of the secant lines through A and D, A and C, and A and B. Then use your calculations to make a conjecture about the slope of the line tangent to the graph of f at x = p>2.

Q: Prove that for any point (a, b) ≠ (0, 0) on the parabola y = x2 , the normal line intersects the…

A: The given equation of parabola is y=x2  For the point (a,b)≠ (0, 0), the slope of normal is and at…

Q: (b) Compute the slope of the line that is parallel to the xz-plane and tangent to the surface z = x…

A: Given: Let's calculate the slope of the line tangent to the surface z=x(x+y) at the point that is…

Q: Find the orthogonal trajectory of y = c1(sec x + tan x)

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A: SINCE YOU HAVE ASKED MULTIPLE QUESTIONS IN SINGLE REQUEST, WE WILL BE ANSWERING ONLY THE FIRST…

Q: verify that the given point is on the curve and findthe lines that are (a) tangent and (b) normal to…

A: We plug x=√3 and y=2 in the given equation So the point is on the curve

Q: Find the exact length of the curve. y = In(sec(x)), Osxs n/4

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Q: If the length of the curve y = 3 x7 from the point (0, 0) to the point (xo, f(xo)) is L = 8 , then…

A: Length of curve

Q: Which of the following are the tangent lines perpendicular to y = x, to the curve x + y² = 8? Oy=x+8…

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Q: Consider the curve in the xy-plane repre- sented by = e' and y = The slope of the line tangent to…

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Q: Show that if P lies on the intersection of the two curves x2 - y2 = c and xy = d (c, d constants),…

A: Given : The two curves x2 - y2 = c and xy = d intersect at point P where c and d are constants.…

Q: Show that if P lies on the intersection of the two curves x2 − y2 = c and xy = d (c, d constants),…

A: Solution:Givenx2-y2=c     and   xy=dTo prove tangent curve at P are…

Q: Let f(x) =- 3x – cos-1(x), then the equation %3D of the normal line to the curve at the point (0,0)…

A:

Q: For the function z = = f(x, y) = x² − xy² + y³, find the equations of the tangent lines in both the…

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Q: Find the exact length af te of the curve, 2. x-

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Q: Consider the curve r (t) = (t, tº, tº) ; t > 0 Does the point (1, 1, 1) belong to the curve and if…

A: Please refer the attached image for complete solution.

Q: 7. Given that the line y = r -b is an asymptote to the ar curve y a,b €R,rz 1, then b = A -1-a 1-a C…

A:

Q: Find the condition that the line x cos a + y ain a =p may touch the curve yh =1. am bm

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Q: Let a(t)-(1+t,t?,2+t?), then the equation of the tangent line at t-1 is: (2+t,-1+2t,3+2t)…

A: Here, a'(t)=(1,2t,2t)a'(1)=(1,2,2)anda(1)=(2,1,3)

Q: 8) Consider the oriented curve C given by C=C₁U C2, where C₁: y = x³, and C₂: y = x², as shown in…

A:

Q: Find the point at which the line f(r) = -z+1intersects the line g(z) = 4x +1 %3D

A: At point of intersection both the functions will have the same value therefore f = g

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A:

Q: Given the function z = = f(x,y) = 2y³ -5xy-3x²+2, find the equation of the tangent plane at (1,-1),…

A: Given problem:- Given the function z  = f(x, y) = 2y³ −5xy-3x²+2, find the equation of the tangent…

Q: find an equation of the tangent plane at the given point. G(u,w) = sin(uw), ( π/6 , 1)

A: Given that: G(u,w)=sin(uw) Now, Find the partial derivatives of Gu(u,w)  and  Gw(u,w) at point π6,1.…

Q: Below is a plot of a cardioid, which consists of points satisfying x² + y² = (2x² + 2y² - x)² 0.6…

A:

Q: Find the length of the curve given by S(x) = (a -) defined on the interval (1, 2].

A: Consider the given curve:

Q: Given that the line y =r- b is an asymptote to the ar curve y a,b €R,r = 1, then b = I-1 -1-a B 1-a…

A:

Q: Mike found the orthogonal line to the curve f (x) =- 3sin( zx) +1 at x=-1. In which step did he…

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Q: Find the exact length of the curve. y = In(sec(x)), 0 <x< a/6

A:

Q: Find the exact length of the curve. y = In(1 - x²), o s x s

A: Solve the problem using definite integral

Q: Find the value of c such that line y=3/2x+6 is tangent to the curve y=cx1/2.

A: Given,  line y=3/2x+6 is tangent to the curve y=cx1/2.

Q: Find the coordinates of the point(s) on the curve 2y = 10 – a² that are closest to the origin. а) О…

A: Given that

Q: C is the curve y = e® from (2, e²) to (0,1) and F = (x², -y), compute ſ, F-dr.

A: Given that, C is the curve y=ex from 2,e2 to 0,1 and F=x2,-y Then, F=x2,-y=x2,-ex So, F→=x2i^-exj^

Q: Let C be any curve.

A:

Q: Find the equation of the tangent plane to z = e9 +x + x³ + 6 at the point (3, 0, 253).

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Q: 2. Find for x? +x+ y? +2 3xy the lines that are tangent and normal to the curve at point (3,2).

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Q: The tangent line to the graph of y = In(x² – 1) is perpendicular to the line a + y = -1 at the point…

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Q: Find the orthogonal trajectory of the given family of curves: 1. y = C(sec x + tan x)

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Q: The Path y= f(t) descibed by a point moving in a Straight line is given by the following table :…

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Q: Find the equation of the tangent plane to the graph of f(x, y) = 8x² – 2xy? at the point (5, 4). A)…

A: We first find the partial derivatives  So the direction vector of the normal is = (48,-80)

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Q: Let f(x) = 3x cos-1(x), then the equation 2 of the normal line to the curve at the point (0, 0) is…

A:

Q: Find the diffrential equation of for each of the curves determined by the condition that at each…

A: length of subtangent =ydydx

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A:

Q: Consider the path r(t) = (4t, 2t², 2 In t) defined for t > 0. Find the length of the curve between…

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Q: 3. Find the length of the curve y = x - In a, 1< x < 4.

A: Note: According to bartleby we have to answer only first question please upload the question…

Q: draw the slope field for y' = x² – y², then use it to sketch three solution curves. %|

A:

Q: Find the exact length of the curve. y = In(1 - x²), o sxs

A: Find dy/dx, then put it in the arc length formula.

Rate of Change

The relation between two quantities which displays how much greater one quantity is than another is called ratio.

Slope

The change in the vertical distances is known as the rise and the change in the horizontal distances is known as the run. So, the rise divided by run is nothing but a slope value. It is calculated with simple algebraic equations as: