# Applications of Calculus

The derivative of a polynomial $$P(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0$$ is given by $P^{\prime}(x)=na_nx^{n-1}+(n-1)a_{n-1}x^{n-2}+\cdots+a_1.$ The inverse operation, the indefinite integral, is given by $\int P(x)dx=\frac{a_n}{n+1}x^{n+1}+\frac{a_{n-1}}nx^n+\cdots+ a_0x+C.$ If the polynomial $$P$$ is not given by its coefficients but rather by its canonical factorization, as $$P(x)=(x-x_1)^{k_1}\cdots(x-x_n)^{k_n}$$, a more suitable expression for the derivative is obtained by using the logarithmic derivative rule or product rule: $P^{\prime}(x)=P(x)\left(\frac{k_1}{x-x_1}+\cdots+ \frac{k_n}{x-x_n}\right).$ A similar formula can be obtained for the second derivative.

Problem 17

Suppose that real numbers $$0=x_0< x_1< \cdots< x_n< x_{n+1}=1$$ satisfy $\sum_{j=0,\,j\neq i}^{n+1}\frac{1}{x_i-x_j}=0\quad\mbox{for } i=1,2,\dots,n.\quad\quad\quad\quad\quad(1)$ Prove that $$x_{n+1-i}=1-x_i$$ for $$i=1,2,\dots,n$$.

What makes derivatives of polynomials especially suitable is their property of preserving multiple zeros.

Theorem 6.1

If $$(x-\alpha)^k\mid P(x)$$, then $$(x-\alpha)^{k-1}\mid P^{\prime}(x)$$.

Problem 18

Determine a real polynomial $$P(x)$$ of degree at most 5 which leaves remainders $$-1$$ and 1 upon division by $$(x-1)^3$$ and $$(x+1)^3$$, respectively.

Problem 19

For polynomials $$P(x)$$ and $$Q(x)$$ and an arbitrary $$k\in\mathbb{C}$$, denote $P_k=\{z\in\mathbb{C}\mid P(z)=k\}\quad\mbox{and}\quad Q_k=\{z\in\mathbb{C}\mid Q(z)=k\}.$ Prove that $$P_0=Q_0$$ and $$P_1=Q_1$$ imply that $$P(x)=Q(x)$$.

Even if $$P$$ has no multiple zeros, certain relations between zeros of $$P$$ and $$P^{\prime}$$ still hold. For example, the following statement holds for all differentiable functions.

Theorem 6.2 (Rolle’s Theorem)

Between every two zeros of a polynomial $$P(x)$$ there is a zero of $$P^{\prime}(x)$$.

Corollary

If all zeros of $$P(x)$$ are real, then so are all zeros of $$P^{\prime}(x)$$.

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