Question Assignment 4.5: The Dimension of a Vector Space Score: \( 54 / 110 \quad 9 / 11 \) answered Prove the following statement. Your work should be legible, and all of your logic should be clear and justified. The space \( \mathbb{P} \) of all polynomials is an infinite-dimensional space. Question Help: \( \square \) Message instructor

(a) Find all polynomials \( f \) that satisty the equation: \( f(x+2)=f(x)+2 \) for every real number \( x \). (b) Find all polynomials \( g \) that satisfy the equation: \[ g(2 x)=2 g(x) \text { for every real number } x . \] (c) The problems above are of the following type: Given functions \( H \) and \( J \), find all polynomials \( Q \) that satisfy the equation: \[ J(Q(x))=Q(H(x)) \text { for every } x \text { in } S \] where \( S \) is a subset of real numbers. In parts (a) and (b), we have \( J=H \) and \( S \) is all real numbers, but other scenarios are also interesting. For example, the choice \( J(x)=1 /(x-1) \) and \( H(x)=1 /(x+1) \), generates the Find all polynomials \( Q \) that satisfy the equation: \[ \frac{1}{Q(x)-1}=Q\left(\frac{1}{x+1}\right) \] for every real number \( x \) such that those denominators are nonzero. Is this one straightforward to solve? (d) Make your own choice for \( J \) and \( H \), formulate the problem, and find a solution. Choose \( J \) and \( H \) to be non-trivial, but still simple enough to allow you to make good progress toward a solution.

< 1 of 1 Figure 120 грm 2.0 m The 300 g bar is rotating as shown in (Figure 1). Part A What is the magnitude of the angular momentum of the bar? ? kg m² /s Submit Request Answer Part B What is the direction of the angular momentum of the bar? O into the page O out of the page

v MI 7. Let (r) = for r e A := (0,0) (i) Find the pointwise limit f off (ii) Show that each f, is bounded on A but the pointwise limit f of the sequence is not bounded on A. (ii) Decide whether or not the convergence in () is uniform, prove your claim. (iv) use the e-strip to prove your claim in (iii).

A solid disk, a hoop, and a solid sphere are released at the same time at the top of an inclined ramp. Each has mass M and radius R. They all roll without slipping. In what order (fastest to slowest) do they reach the bottom of the ramp, if the moments of inertia of the disk, hoop, and sphere are ½ MR2, MR2, and 2/5 MR2? A) disk, hoop, sphere B) disk, sphere, hoop C) sphere, hoop, disk D) hoop, sphere, disk E) sphere, disk, hoop

R W R3 ww 10 12 ww R2 1 Write down the Kirchhoff Junction equation and solve it for I; in terms of 12 and 13. Write the result here: Ij = write down the Kirchhoff Loop equation for a loop that starts at the lower left corner and follows the perimeter of the circuit diagram clockwise. 0 write down the Kirchhoff Loop equation for a loop that starts at the lower left corner and touches the components 10V, Ry, 4V, and Ry O The resistors in the circuit have the following values . - R2 • Ry=60 Ry=130 Solve for all the following (some answers may be negative); 11 - Amperes 12 Amperes 13 Amperes NOTE: Forthin

3. Find the area between the curve y = 2^x andthe chord joining the points (0,1) and (2,4). 4. Find the coordinates of the centroid bounded bythe curve y = x^3 and the line x = 2.