https://docs.google.com/document/d/1oToUlVF-r3SmJcumwfrbUNSoXwojKLST39iNCcK4JDM/edit?hl=en_US
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a) In order to make the function continuous, we need only find a value m so that both halves of the piecewise function give the same y-value when evaluated at the junction x=2 (more precisely, a value such that the two-sided limit exists at x = 2, and agrees with the function value of at least one half of the piecewise function). Set both expressions equal:
x² - mx - 6 = x² + 2
Remove x² from both sides to get:
-mx - 6 = 2
Add 6 to both sides to get:
-mx = 8
It is here that we can substitute that target value of x as 2, and solve for m:
-m(2) = 8
-2m = 8 Divide both sides by (-2)
m = -4
Be aware, the original equation contained a negative already, so substituting the value of m will yield a positive value for the coefficient of x in the function:
f(x) = { x² - (-4)x - 6 for x < 2; x² + 2 for x ≥ 2
f(x) = { x² + 4x - 6 for x < 2; x² + 2 for x ≥ 2
b) Continuity does not necessarily imply differentiability. Thus, even though our function has been made continuous at x = 2, it may or may not be differentiable there. A continuously differentiable function is by definition continuous, and has a continuous function as its derivative. The derivative of a differentiable function never has a "jump" discontinuity. By taking the derivative of each portion of the piecewise function, we can determine if it is continuously differentiable:
f'(x) = { 2x + 4 for x < 2; 2x for x ≥ 2
We see here that the derivative does, in fact, have a jump discontinuity, because the y-values of the two halves of the piecewise derivative are not the same at the junction x=2 (more precisely, there is no two-sided limit of this derivative at x=2). Therefore, the function f(x) is NOT continuously differentiable at x = 2. For a more detailed look at differentiability, take a look here:
x² - mx - 6 = x² + 2
Remove x² from both sides to get:
-mx - 6 = 2
Add 6 to both sides to get:
-mx = 8
It is here that we can substitute that target value of x as 2, and solve for m:
-m(2) = 8
-2m = 8 Divide both sides by (-2)
m = -4
Be aware, the original equation contained a negative already, so substituting the value of m will yield a positive value for the coefficient of x in the function:
f(x) = { x² - (-4)x - 6 for x < 2; x² + 2 for x ≥ 2
f(x) = { x² + 4x - 6 for x < 2; x² + 2 for x ≥ 2
b) Continuity does not necessarily imply differentiability. Thus, even though our function has been made continuous at x = 2, it may or may not be differentiable there. A continuously differentiable function is by definition continuous, and has a continuous function as its derivative. The derivative of a differentiable function never has a "jump" discontinuity. By taking the derivative of each portion of the piecewise function, we can determine if it is continuously differentiable:
f'(x) = { 2x + 4 for x < 2; 2x for x ≥ 2
We see here that the derivative does, in fact, have a jump discontinuity, because the y-values of the two halves of the piecewise derivative are not the same at the junction x=2 (more precisely, there is no two-sided limit of this derivative at x=2). Therefore, the function f(x) is NOT continuously differentiable at x = 2. For a more detailed look at differentiability, take a look here:
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