# calculus

1. Given the curve
a. Find an expression for the slope of the curve at any point (x, y) on the curve.
b. Write an equation for the line tangent to the curve at the point (2, 1)
c. Find the coordinates of all other points on this curve with slope equal to the slope at (2, 1)

1. 👍
2. 👎
3. 👁
4. ℹ️
5. 🚩
1. sorry here's the equation:
x+xy+2y^2=6

1. 👍
2. 👎
3. ℹ️
4. 🚩
2. x+xy+2y^2=6

(a)
x(1+y)=6-2y²
x=(6-2y²)/(1+y)
differentiate with respect to y to get
dx/dy=-4x/(x+1)-(6-2x^2)/(x+1)^2
So
dy/dx
= 1/(dx/dy)
= -(1/2)(x^2+2x+1)/(x^2+2x+3)

(b) For the point (2,1), first check that it lies on the curve.
f'(2)=-(1/2)(2^2+2*2+1)/(2^2+2*2+3)
=-9/22

(c)
Now solve for
f'(x)=-9/22
to get x=-4 or x=2

Verify all the numerical values.

1. 👍
2. 👎
3. ℹ️
4. 🚩
3. There was a mistake in the calculation of dx/dy. It should have been expressed in terms of y and not x.

Given : x+xy+2y^2=6
(a)
x(1+y)=6-2y²
x=(6-2y²)/(1+y)
differentiate with respect to y to get
dx/dy=-4y/(y+1)-(6-2y^2)/(y+1)^2
So
f'(y)=dy/dx
= 1/(dx/dy)
= -(1/2)(y^2+2y+1)/(y^2+2y+3)

(b) For the point (2,1), first check that it lies on the curve.

f'(y)=
f'(1)=-(1/2)(1^2+2*1+1)/(1^2+2*1+3)
=-1/3

Tangent line with slope -1/3 passing through (2,1) is:
(y-1) = (-1/3)(x-2)

(c)
Now solve for
f'(y)=-1/3
to get x=-3 or x=1

Please verify all the numerical values.

1. 👍
2. 👎
3. ℹ️
4. 🚩

## Similar Questions

1. ### Calculus

Consider the curve given by y^2 = 2+xy (a) show that dy/dx= y/(2y-x) (b) Find all points (x,y) on the curve where the line tangent to the curve has slope 1/2. (c) Show that there are now points (x,y) on the curve where the line

2. ### Calculus

Polar Equation Question The figure above shows the graph of the polar curve r=1−2cosθ for 0≤θ≤π and the unit circle r=1. ﻿(a) Find the area of the shaded region in the figure. Question 2 (b) Find the slope of the line

3. ### Calculus

The line that is normal to the curve x^2=2xy-3y^2=0 at(1,1) intersects the curve at what other point? Please help. Thanks in advance. We have x2=2xy - 3y2 = 0 Are there supposed to be 2 equal signs in this expression or is it x2 +

4. ### Maths

a) A curve is defined by the following parametric equations x= 3t/t^2 +1 , y=1/t^2 +1 (i) Find an expression, in terms of t for the gradient of the curve. (ii) Determine the value of the gradient at the point where t=0

1. ### calculus

The point P(7, −4) lies on the curve y = 4/(6 − x). (a) If Q is the point (x, 4/(6 − x)), use your calculator to find the slope mPQ of the secant line PQ (correct to six decimal places) for the following values of x. (i) 6.9

2. ### Calculus

Find normals to the curve xy+2x-y=0 that are parallel to the line 2x+y=0 I have the answer: at(-1,-1), y=-2x-3, and at (3,-3), y=-2x+3 How do we get this? Thanks. xy + 2x - y=0 x dy/dx + y + 2 - dy/dx=0 dy/dx (x-1)= -y-2 dy/dx= -

3. ### Economics

What happens when production is inside the production possibilities curve? It is not possible for the production to move inside the curve. The production is not maximized, so some resources are unused. What does the slope of the

4. ### calc

The slope of the tangent line to a curve at any point (x, y) on the curve is x divided by y. What is the equation of the curve if (3, 1) is a point on the curve?

1. ### Calc AB

Suppose that f(x) is an invertible function (that is, has an inverse function), and that the slope of the tangent line to the curve y = f(x) at the point (2, –4) is –0.2. Then: (Points : 1) A) The slope of the tangent line to

2. ### calculus

The slope of a curve is at the point (x,y) is 4x-3. Find the curve if it is required to pass through the point (1,1). Work... 4(1)-3=1 y-1=1(x-1) y=x

3. ### last calc question, i promise!

given the curve x + xy + 2y^2 = 6... a. find an expression for the slope of the curve. i got (-1-y)/(x + 4y) as my answer. b. write an equation for the line tangent to the curve at the point (2,1). i got y = (-1/3)x + (5/3). but i

4. ### Math

The line has equation y=2x+c and a curve has equation y=8-2x-x^2. 1) for the case where the line is a tangent to the curve, find the value of the constant c. 2) For the case where c = 11, find the x-coordinates of the points of