robertmatthew said:Homework Statement
Evaluate the difference quotient of:
f(1+h)-f(1) / h
if f(x)=x3
and simplify your answer.
Homework Equations
The Attempt at a Solution
I took the derivative of x3 to be 3x2 and solved for when x=1 (given from the difference quotient) and got 3, but when I enter the value it's marked incorrect. Not sure what I'm doing wrong, unless I just interpreted the question the wrong way.
The difference quotient is a mathematical expression used to calculate the slope of a curve at a specific point. It is also known as the average rate of change and is represented by the formula (f(x+h) - f(x)) / h, where h is a small change in the x-value.
The difference quotient is an essential concept in calculus as it is used to find the slope of a curve, which is necessary for calculating derivatives. It is also used to estimate the instantaneous rate of change at a specific point on a curve.
The difference quotient is the general expression used to find the slope of a curve, while the derivative is the specific value of the slope at a given point on the curve. The derivative is calculated by taking the limit of the difference quotient as h approaches 0.
Yes, the difference quotient can be used to find the slope of a line. Since the slope of a line is constant, the difference quotient will always be the same value, regardless of the value of h. The difference quotient for a line can be simplified to (y2-y1)/(x2-x1), which is the familiar slope formula.
The difference quotient has various applications in real life, such as in physics, economics, and engineering. For example, it can be used to calculate the velocity of an object at a specific time, the marginal cost of production, or the rate of change of temperature over time.