Need help with #4 (a) Find the points at which this ellipse crosses the x - axis and show that thetangent lines at these points are parallel(b) Where does the perpendicular line to this ellipse at the point (-1,1) intersectthe ellipse a second time?3. (a) Create a function, f(x) that uses the chain rule and the product rule whenfinding f'(x). Your function should include a trigonometric function.(b) Find the derivative of the function you found in (a)4. Let y = 5- 2x be the tangent line to a function y =is the tangent line to a function y = g(x) + x at x = 3. What is the tangentline to the function y = h(x) at x = 3? Where h(x) = f(x)g(x)+ x²f(x) at x = 3 and y = 2+3%3D5. (a) Find the points on the graph y? = x - 3x +1 where the tangent line ishorizontal.(b) The tangent line to the graph of y + 2xy – y = x³ – 2x² + 2x at the point(0,1) crosses the graph at one other point. Find it analytically.

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Description: Need help with #4 (a) Find the points at which this ellipse crosses the x - axis and show that thetangent lines at these points are parallel(b) Where does the perpendicular line to this ellipse at the point (-1,1) intersectthe ellipse a second time?3

Equation of the tangent line y=f(x) at x=x0 is (y-f(x0))=f'(x0)(x-x0)

Answered: 4. Let y = 5-2x be the tangent line to…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Need help with #4

(a) Find the points at which this ellipse crosses the x - axis and show that the
tangent lines at these points are parallel
(b) Where does the perpendicular line to this ellipse at the point (-1,1) intersect
the ellipse a second time?
3. (a) Create a function, f(x) that uses the chain rule and the product rule when
finding f'(x). Your function should include a trigonometric function.
(b) Find the derivative of the function you found in (a)
4. Let y = 5- 2x be the tangent line to a function y =
is the tangent line to a function y = g(x) + x at x = 3. What is the tangent
line to the function y = h(x) at x = 3? Where h(x) = f(x)g(x)+ x²
f(x) at x = 3 and y = 2+3
%3D
5. (a) Find the points on the graph y? = x - 3x +1 where the tangent line is
horizontal.
(b) The tangent line to the graph of y + 2xy – y = x³ – 2x² + 2x at the point
(0,1) crosses the graph at one other point. Find it analytically.

Expert Solution

Step 1

Equation of the tangent line y=f(x) at x=x₀is

$(y - f (x_{0})) = f' (x_{0}) (x - x_{0})$

Step 2

Given:- y=5-2x is the tangent line to the function y=f(x) at x=3

$S o t h e \tan g e n t l i n e t o t h e f u n c t i o n y = f (x) a t x = 3 i s \Rightarrow (y - f (3)) = f' (3) (x - 3) \Rightarrow y = x f' (3) + (f (3) - 3 f' (3)) C o m p a r i n g w i t h t h e e q u a t i o n o f t h e \tan g e n t l i n e y = 5 - 2 x, w e g e t x f' (3) + (f (3) - 3 f' (3)) = 5 - 2 x \Rightarrow f' (3) = - 2, f (3) - 3 f' (3) = 5 \Rightarrow f (3) = - 1$

Step 3

Given:- y=2+3x is the tangent line to the function y=g(x)+x at x=3

$F o r y = g (x) + x \Rightarrow y' = g' (x) + 1 S o t h e \tan g e n t l i n e t o t h e f u n c t i o n y = g (x) + x a t x = 3 i s \Rightarrow y - (g (x) + x) (3) = (g' (x) + 1)' (3) (x - 3) \Rightarrow y - g (3) - 3 = (g' (3) + 1) (x - 3) \Rightarrow y = (1 + g' (3)) x - 3 (g' (3) + 1) + g (3) + 3 \Rightarrow y = (1 + g' (3)) x + (- 3 g' (3) - 3 + g (3) + 3) \Rightarrow y = (1 + g' (3)) x + (- 3 g' (3) + g (3)) C o m p a r i n g w i t h t h e e q u a t i o n o f t h e \tan g e n t l i n e y = 2 + 3 x, w e g e t (1 + g' (3)) x + (- 3 g' (3) + g (3)) = 2 + 3 x \Rightarrow 1 + g' (3) = 3, - 3 g' (3) + g (3) = 3 \Rightarrow g' (3) = 2 A n d - 3 g' (3) + g (3) = 3 \Rightarrow g (3) = 9$

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