Each of the following statements is false. Provide why the statement is false.
1) Every function that is continuous at x=a is differentiable at x=a.
2) The tangent line to y=f(x) when x=x0 is given by y=fprime(x0)x+f(x0).
3) The linerization of a function f(x) when x=x0 is the value of the derivative of the function when x=x0.
4) If x=f^-1 (y) (meaning y= f(x)), then (dy/dx) = -(f(y))^-2 fprime(y)
you must have some ideas on these. Surely you have read the definition of differentiable. It's late, so I'll have to get back to you, but in the meantime, what are your thoughts? I'm sure your text explains these ideas, and I know google can help.
1) The statement is false because while every differentiable function is continuous, not every continuous function is differentiable. So, there exist functions that are continuous at x=a but not differentiable at x=a.
To see why this is the case, consider the function f(x) = |x| at x=0. This function is continuous at x=0, but it is not differentiable at x=0 because the left and right derivatives do not match.
2) The statement is false because the correct form of the equation for the tangent line to y=f(x) when x=x0 is y=fprime(x0)(x-x0) + f(x0).
To derive this equation, we use the point-slope form of a line, where the slope is given by the derivative of f(x) evaluated at x=x0. The point (x0, f(x0)) lies on the tangent line. Thus, we have the equation y - f(x0) = fprime(x0)(x - x0), which can be rearranged to the correct form.
3) The statement is false because the linearization of a function f(x) when x=x0 is given by f(x) ≈ f(x0) + fprime(x0)(x - x0).
The linearization of a function is an approximation of the function near a specific point, x=x0. It is obtained by adding the tangent line approximation to the function value at x=x0. The derivative of the function at x=x0, fprime(x0), represents the slope of the tangent line at that point.
4) The statement is false because if x = f^(-1)(y) (meaning y = f(x)), then (dy/dx) is equal to 1/fprime(f^(-1)(y)).
The derivative of an inverse function is given by the reciprocal of the derivative of the original function. So, if y = f(x) and x = f^(-1)(y), then (dy/dx) = 1/fprime(f^(-1)(y)).
In the given statement, there is a negative sign and an extra factor of fprime(y)^(-2) which are not correct. The correct derivative is just 1/fprime(f^(-1)(y)).