Question

Let f(x)=12x2. (a) If we wish to find the point Q=(x0,y0) on the graph of f(x)...

Let f(x)=12x2. (a) If we wish to find the point Q=(x0,y0) on the graph of f(x) that is closest to the point P=(4,1), what is the objective function? (Hint: optimize the square of the distance from P to Q.)

Objective function: O(x)=O(x)=____________________________

(b) Find the point Q=(x0,y0)Q=(x0,y0) as described in part (a). Box your final answer.

(c) Verify that the line connecting PP to QQ is perpendicular to the line tangent to f(x)f(x) at QQ. Hint: recall that the lines y=m1x+b1y=m1x+b1 and y=m2x+b2y=m2x+b2 are perpendicular if m1=−1m2m1=−1m2 (assuming neither are horizontal or vertical).

Homework Answers

Know the answer?
Your Answer:

Post as a guest

Your Name:

What's your source?

Earn Coins

Coins can be redeemed for fabulous gifts.

Not the answer you're looking for?
Ask your own homework help question
Similar Questions
Problem: Let y=f(x)be a differentiable function and let P(x0,y0)be a point that is not on the...
Problem: Let y=f(x)be a differentiable function and let P(x0,y0)be a point that is not on the graph of function. Find a point Q on the graph of the function which is at a minimum distance from P. Complete the following steps. Let Q(x,y)be a point on the graph of the function Let D be the square of the distance PQ¯. Find an expression for D, in terms of x. Differentiate D with respect to x and show that f′(x)=−x−x0f(x)−y0 The...
1. Suppose(x0,y0) is the closest point on the line that is the graph of y=5x+7 and...
1. Suppose(x0,y0) is the closest point on the line that is the graph of y=5x+7 and the point (61,0). What is y0? 2. True or False: arctan(2x + 1) is an antiderivative of 1/(2x2+2x+1). 3. Suppose F is an antiderivative of f(x) = 5x4- 3x2 + 1 and F(1) = 5 . What is F(2)? 4. True or False: sinx is an antiderivative of cosx 5. What is the minimum value of 12x2 - x3 on [0,10]?
Let F⃗=xi⃗+(x+y)j⃗+(x−y+z)k⃗ . Let the line l be x=t−1, y=3−4t , z=1−4t . (a) Find a...
Let F⃗=xi⃗+(x+y)j⃗+(x−y+z)k⃗ . Let the line l be x=t−1, y=3−4t , z=1−4t . (a) Find a point P=(x0,y0,z0) where F⃗ is parallel to l . Find a point Q=(x1,y1,z1) at which F⃗ and l are perpendicular. Give an equation for the set of all points at which F⃗ and l are perpendicular.
Let f(x)=22−x2f(x)=22-x2 The slope of the tangent line to the graph of f(x) at the point...
Let f(x)=22−x2f(x)=22-x2 The slope of the tangent line to the graph of f(x) at the point (−4,6) is    . The equation of the tangent line to the graph of f(x) at (-4,6) is y=mx+b for m= and b=   Hint: the slope is given by the derivative at x=−4
Consider the function f(x) = √x and the point P(4,2) on the graph f. a)Graph f...
Consider the function f(x) = √x and the point P(4,2) on the graph f. a)Graph f and the secant lines passing through the point P(4, 2) and Q(x, f(x)) for x-values of 3, 5, and 8. b) Find the slope of each secant line. (Round your answers to three decimal places.) (line passing through Q(3, f(x))) (line passing through Q(5, f(x))) (line passing through Q(8, f(x))) c)Use the results of part (b) to estimate the slope of the tangent line...
find a point of the graph of the function f(x) = e^2x such that the tangent...
find a point of the graph of the function f(x) = e^2x such that the tangent line to the graph at that point passes through the origin. Use a graphing utility to graph f and the tangent line is the same viewing window
Consider the graph of y=f(x)=1−x2  and a typical point P on the graph in the first quadrant....
Consider the graph of y=f(x)=1−x2  and a typical point P on the graph in the first quadrant. The tangent line to the graph at P will determine a right triangle in the first quadrant, as pictured below. a) Find the formula for a function A(x) that computes the area of the triangle through the point P=(x,y)   b) Find the point P so that the area of the triangle is as small as possible: P =()
1. Find a function f given that the slope of the tangent line to the graph...
1. Find a function f given that the slope of the tangent line to the graph of f at any point P(x, y) is given by y' = − 4xy x2 + 1 and the graph of f passes through the point (2, 1). 2. The world population at the beginning of 1980 (t = 0) was 4.5 billion. Assuming that the population continued to grow at the rate of approximately 2%/year, find a function Q(t) that expresses the world...
(1 point) The point P(1/5,10) lies on the curve y=2/x . Let Q be the point...
(1 point) The point P(1/5,10) lies on the curve y=2/x . Let Q be the point (x,2/x). a.) Find the slope of the secant line PQ for the following values of x. If x=1.5/5, the slope of PQ is: If x=1.05/5, the slope of PQ is: If x=0.95/5, the slope of PQ is: If x=0.5/5, the slope of PQ is: Based on the above results, guess the slope of the tangent line to the curve at P(1/5,10)
1.) P(2, 4) lies on the graph of y=x4-2x-4 Q is the point (x, x4-2x-4) Write...
1.) P(2, 4) lies on the graph of y=x4-2x-4 Q is the point (x, x4-2x-4) Write an equation for the slope of the secant line through P and Q as a function of x. 2.) Using the formula above, find the slopes fo 1.9, 1.99, 1.999, 2.001, 2.01, and 2.1 (round 4 decimal places) 3.) What is the slope of the tangent line to the curve y=x4-2x-4 at P(2, 4) Can you please explain in detail how to get the...
ADVERTISEMENT
Need Online Homework Help?

Get Answers For Free
Most questions answered within 1 hours.

Ask a Question
ADVERTISEMENT