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Chapter 1: Problem 4

Find the first derivatives. $$g(y)=y^{2}-2 y+4$$

Short Answer

Expert verified
The first derivative is \(g'(y) = 2y - 2\).

Step by step solution

01

Set up the function

The function provided is \(g(y) = y^2 - 2y + 4\). To find the first derivative, we will differentiate this function with respect to \(y\).
02

Differentiate each term

Differentiate each term of the function. \(y^2\) becomes \(2y \), \(-2y \) becomes \(-2 \), and \(4\) becomes \(0\) since the derivative of a constant is zero.
03

Combine the derivatives

Combine the derivatives of each term: \[ g'(y) = 2y - 2 + 0 = 2y - 2 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

First Derivatives
The concept of the first derivative is fundamental in calculus. A first derivative provides the rate of change of a function with respect to one of its variables. For instance, if you have the function \( g(y) = y^2 - 2y + 4 \), the first derivative, denoted as \( g'(y) \), tells us how \( g(y) \) changes as \( y \) changes.

This is helpful for understanding the behavior of functions, such as finding slopes of curves at any given point.

In this example, computing the first derivative helps indicate the behavior of the polynomial function around different points.
Differentiation
Differentiation is the process of finding a derivative. It involves applying rules that help simplify the computation of how a function changes.

For the function \( g(y) = y^2 - 2y + 4 \), differentiating it term by term is straightforward:
  • Differentiating \( y^2 \) gives \( 2y \)
  • Differentiating \( -2y \) yields \( -2 \)
  • The constant \( 4 \) differentiates to \( 0 \)

Combined, the differentiation process results in the first derivative: \( g'(y) = 2y - 2 \).

This systematic approach ensures you accurately find how a function behaves.
Polynomial Functions
Polynomial functions are expressions that involve sums of powers of a variable. They often appear in the form \( a_n \, y^n + a_{n-1} \, y^{n-1} + ... + a_1 \, y + a_0 \).

Our example, \( g(y) = y^2 - 2y + 4 \), is a polynomial of degree 2, where:
  • \( y^2 \) is the quadratic term
  • \( -2y \) is the linear term
  • \( 4 \) is the constant term

When differentiating polynomial functions, each term鈥檚 differentiation follows rules:
  • The derivative of \( y^n \) is \( n \, y^{n-1} \)
  • A linear term differentiates to the coefficient of the variable
  • A constant term differentiates to \( 0 \)

This simplicity makes polynomial functions easier to differentiate compared to more complex functions.
Calculus Solutions
When solving calculus problems, understanding each step is critical. The example function \( g(y) = y^2 - 2y + 4 \) is differentiated by following a clear process:

  • Identify the function to differentiate.
  • Apply differentiation rules to each term.
  • Combine the derivatives to form the final result.

Following this method ensures clarity in complex problems. For \( g(y) \), this results in: \( g'(y) = 2y - 2 \).

Always remember: proper differentiation yields valuable insights, like finding slopes and optimizing functions.

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Most popular questions from this chapter

(a) Let \(A(x)\) denote the number (in hundreds) of computers sold when \(x\) thousand dollars is spent on advertising. Represent the following statement by equations involving \(A\) or \(A^{\prime}:\) When \(\$ 8000\) is spent on advertising, the number of computers sold is 1200 and is rising at the rate of 50 computers for each \(\$ 1000\) spent on advertising. (b) Estimate the number of computers that will be sold if \(\$ 9000\) is spent on advertising.

Determine whether each of the following functions is continuous and/or differentiable at \(x=1.\) $$f(x)=\left\\{\begin{array}{ll} x & \text { for } x \neq 1 \\ 2 & \text { for } x=1 \end{array}\right.$$

Using the sum rule and the constant-multiple rule, show that for any functions \(f(x)\) and \(g(x).\) $$\frac{d}{d x}[f(x)-g(x)]=\frac{d}{d x} f(x)-\frac{d}{d x} g(x).$$

Apply the three-step method to compute the derivative of the given function. $$f(x)=-x^{2}$$

Determine whether each of the following functions is continuous and/or differentiable at \(x=1.\) $$f(x)=\left\\{\begin{array}{ll} x^{3} & \text { for } 0 \leq x<1 \\ x & \text { for } 1 \leq x \leq 2 \end{array}\right.$$
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