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2. We say that two curves intersect orthogonally if they intersect and their tangent lines are orthogonal at each point in the intersection. For example, the curve y = 0 intersects the curve x2 + y2-...
2. We say that two curves intersect orthogonally if they intersect and their tangent lines are orthogonal at each point in the intersection. For example, the curve y = 0 intersects the curve x2 + y2-1 orthogonally at (-1,0) and (1,0). Let H be the set of curves y2-2.2-b with b є R. (a) Prove that the tangent line of each curve in H at a point (x, y) with y 0 has slope - (b) Let y-f(x) be a...
Each of these problems (Problems 1-4) is worth four points Definition: Two lines or curves are said to be normal to each other at their point of intersection if they intersect there at right angles or, equivalently, if their tangent lines at the point of intersection are 1. A well-known theorem in geometry states that a line which is tangent to a circle is perpendicular to the radius of the circle at the point of tangency. Use implicit differentiation to...
Consider the following. z = x2 + y2, z = 36 − y, (6, -1, 37) (a) Find symmetric equations of the tangent line to the curve of intersection of the surfaces at the given point. x − 6 12 = y + 1 −2 = z − 37 −1 x − 6 1 = y + 1 12 = z − 37 −12 x − 6 = y + 1 = z − 37 x − 6 12 =...
6. (a) Newton's method for approximating a root of an equation f(x) 0 (see Section 3.8) can be adapted to approximating a solution of a system of equations f(x, y) 0 and gx, y) 0. The surfaces z f(x, y) and z g(x, y) intersect in a curve that intersects the xy-plane at the point (r, s), which is the solution of the system. If an initial approxi- mation (xi, yı) is close to this point, then the tangent planes...
The given point is on the curve. Find the lines that are (a) tangent and (b) normal to the curve at the given point. 4x2 + 3xy + 3y2 +17y - 4 = 0,(-1,0) (a) Give the equation of the line that is tangent to the curve at the given point y = (b) Give the equation of the line that is normal to the curve at the given point. y = Suppose that fis an odd function of x....
Consider the parabola y = 7x - x2. Find the slope m of the tangent line to the parabola at the point (1, 6). using this definition: The tangent line to the curve y = f(x) at the point P(a, f(a)) is the line through P with slope m=lim x rightarrow a f(x)-f(a)/x-a provided that this limit exists. m = using this equation: m=lim h rightarrow 0 f(a+h)-f(a)/h m= Find an equation of the tangent line in part (a). y...
Suppose is a closed curve in the plane and that -Y dr + 2? + y2 2 dy = 671 z? + y2 How many self-intersection points must have, at least? By "self-intersection point", I mean a point where the curve intersects itself other than its endpoints. For example, a simple closed curve has zero self-intersection points, and a figure 8 has one self-intersection point. Hint: If a curve has self-intersection points, then it can be divided up into a...
Question (2): (5 Marks) x-1-3-y x-1-6-y:+2 are (A) Determine intersecting or skew. If they intersect, find the point of intersection Given SI: x2-2y2 = 4z2-252 &s2: (0 Show that the tangent planes to the two surfaces at P(2,0,-8) are perpendicular. whether the lines parallel, 2-z & 12 Marks] 4x2 +9y2-24. (B) Find the points on Si at which the tangent plane is parallel to the plane x+y+32-5 3 Marks] Question (2): (5 Marks) x-1-3-y x-1-6-y:+2 are (A) Determine intersecting or...
Math 32-_ Multivariable Calculus HW 3 (1) Consider the two straight lines L1 : (2-t, 3 + 2t,-t) and L2 : <t,-2 + t, 7-20 a) Verify that L1 and L2 intersect, and find their point of intersection. (b) Find the equation of the plane containing L1 and L2 (2) Consider the set of all points (a, y, z) satisfying the equation 2-y2+220. Find their intersection 0 and 2-0. Use that information to sketch a with the planes y =-3,-2,-1,0,...
3. [10] (quadrifolium) Let (a2 + y2) = (2 -)2 be a curve. Find the points on the curve where the normal line is parallel to y 0. re2y, find the normal line at 4. [4] Let (1,0). [0, 10] with f(0) f(10) 0 and 5. 5 Let f(a) be continuous and differentiable on f(5) 4. Mark TRUE or FALSE for the following statements and JUSTIFY. (No points will be given without the correct justification) (A) There is some c...