1. Consider the piecewise function S(1+ x)* – 1, x < 0 g(x) = sin (4x), x > 0 Discover whether g(x) is differentiable at r = 0 using the differentiability test. Without carrying out any further calculation, what can you conclude about the continuity at point x = 0? 2. The second derivative of a parametric function can be calculated using the formula:

Q: 2. Find dy/dx (a) x = 5 sin 16t + 126 and y = 5 cos 16t + 13 (use parametric differentiation) (b)…

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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: 9. Find the constants a, b and c of cubic function f (x) = ax³ + bx + c such that : • The slope of…

A: Given that, f(x) = ax3 + bx  + c Slop of tangent line at x = 3 is 0. f'(1) = 24 f(x) croses the x…

Q: Let y = tan find the differential dy of the function.

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Q: Q2) find the slop for the tangent of the function f (x) sin(xcos(x)) at x = 2n. %3D

A: To find the slope of the tangent of the function fx=sinxcosx at x=2π. Solution: The slope of a…

Q: 4 - Find the direction A in which the function f(x, y) = x cos y + y sin x decreases most rapidly at…

A: We have to find the direction A in which the given function decreases most rapidly.

Q: Determine the equation of the curve whose tangent line at any point (x, y) is e*- cos x and passes…

A: The curve whose tangent line at any point x,y is ex-cos x and passes through 0,4

Q: Let f(x)= 9 sin x/ 2 sin x + 4 cos x Then f'(x)= The equation of the tangent line to y=f(x) at a =…

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Q: - Sva cos 2t dt from x= 0 to x= Find the length of the curve y =

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Q: Show that the function below is contimuous but not differential atr= 0. r sin() if f(r) = 0. if r =…

A: Calculate RHL and LHL

Q: dz Find dz and by implicit differentiation, and confirm that the ду results obtained agree with…

A: Given that,  yex-9sin4z=4z, Now, yex-9sin4z=4z yex-9sin4z-4z=0 ⇒f(x,y,z)=0 where ,…

Q: Find the derivative of y with respect to t at t=2 if a) у %3D х + sinх - cos(1 + x+x?) аnd x = (t/t…

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Q: Given that y is a function of x defined implicitly by the equation: siny = 4x + xy dy Find the first…

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Q: Find the derivative of y with respect to 0. y = In (8 cos 2e) dy %3D de

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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: 1. Find the derivative (dy/dx) of the following functions: In a) y = 3x7- 7x³ +21x2 In b) y = (x2 +…

A: As per the guidelines, we are supposed to answer only one question at once. Kindly repost the…

Q: 3. For the function h(x, y) = x2 + y2 - xy+x find h and h, and then find the tangent plane to h(x,y)…

A: To solve the following problem, We use the standard formula of differentiation and  after that we…

Q: Given: f(x) = / (2 – 4t) sin() - dt %3D tan(2r) 17 – cos(2nt) Find the equation of the line tangent…

A: The tangent is a straight line to a curve that touches the curve at exactly one point. The point…

Q: cal linear approximation of f(x) = √ (1+x) at x0 = 0. Use your approximation formula to approximate…

A: The given function is f(x)=1+x at x0=0 we have to Find the local linear approximation.

Q: 7. Consider the function f: R → R³ given by: t+1 f(t) = sin(cos(t)), tet) t2 +4° Find f'(t) for all…

A: The derivative of the function is the measure of how fast the function is varying with respect to…

Q: what is of point Fyl-2,0) fexsy)=(x3ーy+g)s

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Q: I Find the second derivative(y") of the following functions and simplify the final results. 1. y =…

A: Note : As per our guidelines we are supposed to answer only one question if there are multiple…

Q: a) The tangent line for y = f(z) at the point (c, f(e)) has as its equation y = f(c) + f'(e)(r – e).…

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Q: 2. Consider the squiggly graph of all points (r, y) that satisfy sin(re") = a + y. 1 Using implicit…

A: Given, sinxey=x+y

Q: Q-1: find the derivative of y with respect to the given indepe 1. y = In(Inx) 2. y = In(In(Inx)) x…

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Q: Let h (x) = x sin . 1. Find the x-values where h has horizontal tangent lines in the interval 0, 27.…

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Q: Let  y = f(x) = x ⋅ cos(πx) Find the equation of the line tangent to the graph of  y = f (x) when…

A: Given Let  y = f(x) = x ⋅ cos(πx) equation of the line tangent to the graph of  y = f (x) when x =…

Q: 8. Find the length of the curve defined by x = cos(2t), y = sin(2t) from t = 0 to t = n.

A: If x = f(t), y = g(t) in (a, b) then length of the curve is,

Q: 1. Consider the piecewise function S(1 + x)* – 1, x0 0 using the differentiability test. Discover…

A: First we check continuity then apply definition of Differtiability.

Q: Assuming 9x° – 5y - 4xy = 0 defines y as a differentiable function of x, use the theorem dx - to…

A: Given: 9x3-5y2-4xy=0 We have to calculate dydx(1,1) by the given formula.

Q: The line y = x - 1 is the tangent line to the graph of the function (A)f(x) = sinh x- cos 3x at the…

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Q: 1. Find the first derivative of y with respect to x (y') of the given functi 1. x³y-xy² +x² +y? = 0…

A: To find the first derivative of y using the given equation.

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: Q6:- Show that R = (sin t)i + (cos t)j + v3k has a constant length. And R(t) is orthogonal to its…

A: “Since you have asked multiple questions, we will solve the first question for you. If you want any…

Q: Find the value of the derivative of g (x, y) when we restrict to the curve defined by x = cos t and…

A: Differrtiate with respect to t

Q: Find an equation for the tangent line to the graph of f(x) = -2x2 + 2x + 3 at the point where x = 1…

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Q: (a)Find the equation of the tangent line to the curve 8x'y - 5x? + 7y° = 10(x-3y), at the point…

A: See the attachment

Q: Let f(x,y) = 8x + 10y with x(t) = cos t, and y(t) = sin t. Find the derivative of w = f(x,y) with…

A: Here we have to find the derivative of w with respect to t.

Q: The graph of the function f(x) = sin(x+ 8) is shown along with the points (– 8, f( – 8)) and (– 7,…

A: (1) f(x)=sin(x+8)f'(x)=cos(x+8)slope =f'(-8)=cos(0)=1equation of tangent at…

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: Assuming that the equations define y as a differentiablefunction of x, use Theorem 8 to find the…

A: It is given that,xey+sinxy+y-ln2=0.We have to find the value of dydx at the point 0,ln2.

Q: Suppose z is a differentiable function of x and y implicitly defined by the equation E : 2³ + xe² =…

A: Given function z3+xez=2-y3+lny3 cos(y+1)

Q: Find the third order derivative of y = In(cos(2x)). A. y' = -40sec? (2x)tan(2x) B. y' = -80sec?…

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Q: Find the actual error when the first derivative of f(x) = sin(2x) at *=1/8 s approximated by the…

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Q: 2. If g(x) = x + 2 cos x, then the equation of the tangent line to the graph of g at x = is A. 2y +…

A: Wr have to find the derivative of the given function first.

Q: 2. If g(x) = x + 2 cos x, then the equation of the tangent line to the graph of g at x = is A. 2y +…

A: Equation of tangent line through A(x1 , y1) is given by :- y – y1 = slope x (x – x1)

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: 7) Let x = t – sin(t) , y = 1– cos(t). Determine the equation of the line tangent at t = n.

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