Math Problem Statement

Let y = x^2 + 3. (a) Find the average rate of change of y over intervals [2, 3], [2, 2.5], and [2, 2.1]. (b) Find the instantaneous rate of change of y at x = 2.

Solution

Let's solve this problem step by step.

Given equation:

y=x2+3y = x^2 + 3

(a) Average rate of change of yy over the intervals

The average rate of change of a function y=f(x)y = f(x) over an interval [x1,x2][x_1, x_2] is given by:

Average rate of change=f(x2)f(x1)x2x1\text{Average rate of change} = \frac{f(x_2) - f(x_1)}{x_2 - x_1}

Interval 2, 3:

  1. f(2)=22+3=4+3=7f(2) = 2^2 + 3 = 4 + 3 = 7
  2. f(3)=32+3=9+3=12f(3) = 3^2 + 3 = 9 + 3 = 12

Average rate of change=12732=51=5\text{Average rate of change} = \frac{12 - 7}{3 - 2} = \frac{5}{1} = 5

Interval 2, 2.5:

  1. f(2.5)=(2.5)2+3=6.25+3=9.25f(2.5) = (2.5)^2 + 3 = 6.25 + 3 = 9.25

Average rate of change=9.2572.52=2.250.5=4.5\text{Average rate of change} = \frac{9.25 - 7}{2.5 - 2} = \frac{2.25}{0.5} = 4.5

Interval 2, 2.1:

  1. f(2.1)=(2.1)2+3=4.41+3=7.41f(2.1) = (2.1)^2 + 3 = 4.41 + 3 = 7.41

Average rate of change=7.4172.12=0.410.1=4.1\text{Average rate of change} = \frac{7.41 - 7}{2.1 - 2} = \frac{0.41}{0.1} = 4.1

(b) Instantaneous rate of change at x=2x = 2

The instantaneous rate of change is the derivative of yy with respect to xx. Given y=x2+3y = x^2 + 3, the derivative is:

dydx=2x\frac{dy}{dx} = 2x

At x=2x = 2:

dydxx=2=2(2)=4\frac{dy}{dx}\bigg|_{x=2} = 2(2) = 4

Final answers:

  • (a) Average rates of change:
  • (b) Instantaneous rate of change at x=2x = 2: 4

Would you like more details or have any questions? Here are five related questions for practice:

  1. How do you compute the average rate of change for a cubic function over an interval?
  2. What is the difference between average and instantaneous rates of change?
  3. How would the instantaneous rate of change change if the function were y=x3+3y = x^3 + 3?
  4. What is the meaning of the instantaneous rate of change in real-world scenarios?
  5. How can you approximate the instantaneous rate of change using average rates?

Tip: The instantaneous rate of change at a point is essentially the slope of the tangent line to the function at that point.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Algebra
Calculus
Rate of Change

Formulas

Average rate of change: (f(x2) - f(x1)) / (x2 - x1)
Instantaneous rate of change: Derivative of the function

Theorems

Derivative of a quadratic function

Suitable Grade Level

Grades 11-12, Early College