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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