Question

Finding \(f\) from \(f^{\prime}\) Let $$f^{\prime}(x)=3 x^{2}$$ (a) Compute the derivatives of \(g(x)=x^{3}, h(x)=x^{3}-2,\) and \(t(x)=x^{3}+3 .\) (b) Graph the numerical derivatives of \(g, h,\) and \(t\) (c) Describe a family of functions, \(f(x),\) that have the property that \(f^{\prime}(x)=3 x^{2}\) . (d) Is there a function \(f\) such that \(f^{\prime}(x)=3 x^{2}\) and \(f(0)=0 ?\) If so, what is it? (e) Is there a function \(f\) such that \(f^{\prime}(x)=3 x^{2}\) and \(f(0)=3 ?\) If so, what is it?

Short answer

The family of functions that have the property \(f^{\prime}(x)=3x^2\) is \(f(x) = x^3 + C\) where \(C\) is a constant. The function \(f(x)\) satisfying \(f^{\prime}(x)=3x^2\) and \(f(0)=0\) is \(f(x) = x^3\). The function \(f(x)\) satisfying \(f^{\prime}(x)=3x^2\) and \(f(0)=3\) is \(f(x) = x^3 + 3\).

Step-by-step solution

Calculate Derivatives

To get the derivatives of the given functions \(g(x), h(x), t(x)\), apply the power rule for differentiation, which states that the derivative of \(x^n\) is \(n \cdot x^{n-1}\). \n So, \(g^{\prime}(x) = 3x^2\), \(h^{\prime}(x) = 3x^2\) since the derivative of a constant is zero, and \(t^{\prime}(x) = 3x^2\) since again the derivative of a constant is zero.

Graph the Numerical Derivatives

Here we note that all functions \(g, h, t\) have the same derivative, which is equal to \(3x^2\). Hence, when graphing these derivatives, we would obtain the same parabolic curve opening upwards. This is due to the nature of the derivative of these functions, which are all quadratic in nature (i.e. of the form \(ax^2 + bx + c\)).

A Family of Functions Having \(f^{\prime}(x)=3x^2\)

A family of functions, \(f(x)\), that satisfy this condition can be obtained by integrating \(3x^2\), which gives \(x^3 + C\) where \(C\) is the constant of integration. This \(C\) can be any real number, and leads to a family of functions, as each choice of \(C\) gives a different function \(f(x)\).

Function \(f\) such that \(f^{\prime}(x)=3x^2\) and \(f(0)=0\)

If we desire a function \(f(x)\) such that \(f(0) = 0\), we would choose \(C = 0\) from our family of functions obtained in Step 3. So \(f(x) = x^3\) would be the desired function.

Function \(f\) such that \(f^{\prime}(x)=3x^2\) and \(f(0)=3\)

Lastly, to find a function \(f(x)\) such that \(f(0) = 3\), we would require to set \(C = 3\) in our family of functions from Step 3. Therefore, the function would be \(f(x) = x^3 + 3\)