[카이스트 방학숙제2] winter 2022 assignment 2 [더플러스수학]

 

Problem 1

Suppose that a function \(\displaystyle f : \mathbb{R} \rightarrow \mathbb{R}\) satisfies the following conditions for all real values \(\displaystyle x\) and \(\displaystyle y\):

(i) \(\displaystyle f(x + y) = f(x) · f(y)\).

(ii) \(\displaystyle f(x) = 1 + xg(x)\), where \(\displaystyle \lim\limits_{x \rightarrow 0} g(x) = 1\)

Show that the derivative f′(x) exists at every value of x (that is, f(x) is differentiable) and that
\(\displaystyle f′(x) = f(x)\).

Problem 2

(a) Find the equation for the tangent line at the point \(\displaystyle (1, ~1)\) to the curve given by the equation

\(\displaystyle y^2 (2 − x) = x^3 \).

(b) Give an argument that proves the equality

\(\displaystyle \cos \left( \sin^{−1}(x) \right)= \sqrt{1-x^2}\)

(c) Assume that \(\displaystyle y = \sin^{−1}(x)\) is a differentiable function of \(\displaystyle x\). Use implicit differentiation on the equation \(\displaystyle x = \sin y\) to show that

\(\displaystyle \frac{dy}{dx}= \frac{1}{\sqrt{1-x^2 }}\).