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Calc 221 10/21 Maximize the Area of a rectangle that is inside the curve y=x² A= x(1-y) x(1-x²) 0 ? x ? 1 Da/dx = 1-3x² Set equal to 0 0= 1-3x² X= ?? Maximum area occurs when x= ?? and the maximum area is 2/3(??) Is there a max and min to the line y= 1/x + x²? X > 0 Dy/dx = -1/x² + 2x -1/x² +2x = 0 X= ³?½ There is no maximum Minimum = 3?½ Battery Problem: P = rI² = rV²/(R+r)² P = rV²(R+r)-² dP/dr = V²[(R+r)-² + r(-2)(R+r)-³] Set = to 0 2r/(R+r)³ = 1/(R+r)² Multiply both sides by (R+r)³ 2r= R+r R=r The maximum is when the internal resistance (R) equals the external resistance (r). Minimize Distance = Water/fire problem S = ? 4 + x² + ? 9 + (5-x)² 0 ? x ? 5 Ds/dx = ½ (4+x²) ? ½ (2x) + ½ (9+(5-x)²)- ½ (2)(5-x)(-1) Set = to 0 5-x/?9+(5-x)² = x/?4+x² Cross multiply (5-x)(?4+x²) = x(?9+(5-x)²) 4(5-x)² = 9x² 4(25-10x+x²) = 9x² 5x²+40x-100 x²+8x-20 (x-2)(x+10) When x=2 The minimum always occurs at place where two angles are equal

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