Question

1. Find the point on the curve y = x2 that is closest to (0, 5).

2. Find the function f(x),iff′′(x)=sinx+x and f(0)=f(π)=0.

3. Find derivatives of the following functions. a) arcsin( square root 3x)

Answer #1

1.

y = x2

Let the point on the curve be (a,b)

As this point lies on the curve, it will satisfy the equation of the curve. Therefore,

b = a^2

Therefore, the point is (a, a2)

The slope of the line joining the point on the curve and the point (0,5) is

**Slope** = (y2-y1)/(x2-x1) = (a^2 - 5)/(a - 0) =
**(a^2-5)/a**

Slope at any point on the curve = dy/dx = 2x

At the point (a,a^2), **Slope = 2a**

We know that the line connecting the point on the curve and the point (0,5) will be perpendicular to the tangent at the point on the curve.

Therefore, the product of the two slopes will be -1.

Therefore,

((a^2-5)/a)*2a = -1

(a^2-5)*2 = -1

(a^2-5) = -1/2

(a^2) = (-1/2)+5 = 9/2

**a = sqrt(9/2)**

**a = +3/sqrt(2) and a = -3/sqrt(2)**

Therefore, there are 2 points which are closest to the point.

**(3/sqrt(2), 9/2) and (3/sqrt(2),9/2)**

Find the point on the line 3x + y = 5 that is closest to the
point (−4, 2). (x, y) =?

Find the point on the graph of the function closest to the given
point (1, 5)
F(x)=x2-1

Find the point on the curve y = sqrt(x) is closest to the point (3,
0) , and find the value of this minimum distance.
Use the Second Derivative Test to show that this value is a
minimum. Show work

1)Consider the curve y = x + 1/x − 1 .
(a) Find y' .
(b) Use your answer to part (a) to find the points on the curve
y = x + 1/x − 1 where the tangent line is parallel to the line y =
− 1/2 x + 5
2) (a) Consider lim h→0 tan^2 (π/3 + h) − 3/h This limit
represents the derivative, f'(a), of some function f at some number
a. State such an...

In xy-plane, find the point on the curve y^2/x - 9/x = 1
closest to the origin.
a. Name the function f which you are minimizing, and name your
constraint g.
b. Set up a LaGrange multiplier equation, and system of n
equations, n unknowns. Then solve the system of equations.

Find the point on the curve y=4x+2 closest to
the point (0,4).
(x,y)= (_____,_____)

Let y be the solution of the equation
a)
y ′ = 2 x y, satisfying the condition y ( 0 ) = 1.
Find the value of the function f ( x ) = ln ( y ( x ) )
at the point x = 2.
b)
Let y be the solution of the equation
y ′ = sqrt(1 − y^2) satisfying the condition y ( 0 )
= 0.
Find the value of the function f ( x...

Find the point on the line 3x + y = 7 that is closest to the
point (−4, 3)
(x,y) =

a) Let T be the plane 3x-y-2z=9. Find the shortest distance d
from the point P0 = (5, 2, 1) to T, and the point Q in T that is
closest to P0. Use the square root symbol where needed.
d=
Q= ( , , )
b) Find all values of X so that the triangle with vertices A =
(X, 4, 2), B = (3, 2, 0) and C = (2, 0 , -2) has area (5/2).

Find the point on the curve y = sqrt (x) that is closest to the
point (5,0). Please type in only the exact values for x and y,
separated by a comma. You should not enter any parentheses as they
are already included below. (x, y) =

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