Non-Differentiability Corner Illustrate that the following function f(x) is not differentiable at x-2 due to a...

 

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Non-Differentiability Corner Illustrate that the following function f(x) is not differentiable at x-2 due to a corner. - x2 + 4x +1 x<2 f(x) x2 - 2x+5 x22 { f(x) = { -x^2 + 4x + 1, x<2 (x^2 - 2x + 5, x>= 2 a. Evaluate the right-sided limit b. Evaluate the left-sided limit C. Exaplain why the function is non-differentiable at x=2 and how we know this is due to a corner. Infinite limit - cusp or vertical tangent a) Consider the function f(x) defined as follows f(x) = x { fx) = cuberoot(x) } Find the derivative f(X) by the power rule and show f(x) is not defined at x-0. [Note that this case where lim f'(x) = +0 and lim f(x) = +0 x+ 0+ corresponds to a vertical tangent line.] b) Consider the function g(x) defined as follows olx) = x2 { g(x) = cuberoot(x^2) } Find the derivative g'(x) by the power rule and show g'(x) is not defined at x=0. [Note that this case where lim g'(x) = + 00 and lim g'(x) = - corresponds to a cusp (which also has infinite slope-and a change in sign).] Non-Differentiability Corner Illustrate that the following function f(x) is not differentiable at x-2 due to a corner. - x2 + 4x +1 x<2 f(x) x2 - 2x+5 x22 { f(x) = { -x^2 + 4x + 1, x<2 (x^2 - 2x + 5, x>= 2 a. Evaluate the right-sided limit b. Evaluate the left-sided limit C. Exaplain why the function is non-differentiable at x=2 and how we know this is due to a corner. Infinite limit - cusp or vertical tangent a) Consider the function f(x) defined as follows f(x) = x { fx) = cuberoot(x) } Find the derivative f(X) by the power rule and show f(x) is not defined at x-0. [Note that this case where lim f'(x) = +0 and lim f(x) = +0 x+ 0+ corresponds to a vertical tangent line.] b) Consider the function g(x) defined as follows olx) = x2 { g(x) = cuberoot(x^2) } Find the derivative g'(x) by the power rule and show g'(x) is not defined at x=0. [Note that this case where lim g'(x) = + 00 and lim g'(x) = - corresponds to a cusp (which also has infinite slope-and a change in sign).]