[Math] Solve for constants: Derivatives using first principles

calculusderivativeslimits

  • Question

    Find the values of the constants $a$ and $b$ such that $$\lim_{x \to 0}\frac{\sqrt[3]{ax + b}-2}{x} = \frac{5}{12}$$

  • My approach

    • Using the definition of the derivative, $$f'(x) = \lim_{h \to 0} \frac{f(x+h) – f(x)}{h}$$

    • I view limit as a derivative of a function $f$ at some value, let's call that value $c$, as follows :$$f'(c) = \lim_{x\to 0}\frac{f(c + x) – f(c)}{x} = \frac{\sqrt[3]{ax + b} – 2}{x} = \frac{5}{12}$$

    • Now I deduce the following: $$f(c + x) = \sqrt[3]{ax + b}$$ and $$f(c) = 2$$

    • Use limits as follows: $$\lim_{x\to 0} f(c + x) = f(c) = 2$$ that is, $$\lim_{x\to 0} f(c + x) = \lim_{x\to 0} \sqrt[3]{ax + b} = \sqrt[3]{b} = 2$$ now solve for $b$, $$\sqrt[3]{b} = 2 \Leftrightarrow b = 8$$

    • Since I know that $$f(c + x) = \sqrt[3]{ax + 8}$$, I can solve for $a$, which is $$a = \frac{[f(c+x)]^3 – 8}{x} = \frac{[f(c+x)]^3 – [f(c)]^3}{x}$$

    • Let $g(x) = [f(x)]^3$, such that $$g'(x) = 3\cdot [f(x)]^2 \cdot f'(x)$$
      so $$g'(c) = 3\cdot [f(c)]^2 \cdot f'(c) = 3 \cdot 4 \cdot \frac{5}{12} = 5$$

    • Rephrase $g'(c)$ using first principles such that $$g'(c) = \lim_{x \to 0}\frac{g(c + x)- g(c)}{x}= \lim_{x \to 0}\frac{[f(c + x)]^3 – [f(c)]^3}{x} = \lim_{x \to 0} a = 5$$

    • Since $a$ is a constant, $\lim_{x \to 0} a = a$, that is, $$a = 5$$

    • My solution: $b = 8, a = 5$.

Please have a look at my approach and give me any hints/suggestions regarding the solution and/or steps taken.

Best Answer

It is fine. However, there is a short proof if you know this

$(1+x)^\frac1n=1+\frac1nx+o(x)$, where $\lim_{x\to 0}\frac{o(x)}{x}=0.$

So $$\lim_{x \to 0}\frac{\sqrt[3]{ax + b}-2}{x} = \lim_{x \to 0}\frac{b^\frac13 \sqrt[3]{\frac a bx + 1}-2}{x}=\lim_{x \to 0}\frac{(b^\frac13-2)+ \frac13 b^\frac13 \frac abx+b^\frac13o(x)}{x}=\frac{5}{12}.$$

So, $b^\frac13-2=0$ and $\frac13 b^\frac13 \frac ab=\frac{5}{12}$.

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