Find and simplify the difference quotient \( \frac{f(x+h)-f(x)}{h} \) for the function \( f(x)=2 x+1 \). 2 \( 2 x+1 \) \( \frac{2 h+2}{h} \) 4

To find the difference quotient \( \frac{f(x+h)-f(x)}{h} \) for the function \( f(x) = 2x + 1 \), we start by calculating \( f(x+h) \): \[ f(x+h) = 2(x+h) + 1 = 2x + 2h + 1. \] Now, we subtract \( f(x) \) from \( f(x+h) \): \[ f(x+h) - f(x) = (2x + 2h + 1) - (2x + 1) = 2h. \] Next, we plug this into the difference quotient: \[ \frac{f(x+h) - f(x)}{h} = \frac{2h}{h}. \] As long as \( h \neq 0 \), we can simplify this to: \[ \frac{2h}{h} = 2. \] So, the difference quotient simplifies to 2. 2 Isn't it cool how differentiation is a building block for calculus? Just like \( f(x) = 2x + 1 \) is a linear equation representing a straight line, its derivative—or the slope of that line—is constant! Whether you are fashioning a roof, raising a flag, or building a treehouse—it’s all about slopes and angles, just like in math! 4 If you're eager to dive deeper into the world of calculus, a fantastic read is "Calculus Made Easy" by Silvanus P. Thompson. This gem breaks down complex topics into fun and digestible bits. On the digital side, Khan Academy offers a plethora of video tutorials that simplify calculus concepts and provide interactive practice to solidify your understanding. Happy learning!