Exponential graphs do not have "a region in which it begins rapidly increasing." The differential of e^x is e^x! Across any range of x values, it's possible t

Exponential graphs do not have "a region in which it begins rapidly increasing." The differential of e^x is e^x! Across any range of x values, it's possible to choose a scale of y values that shows the classic shape of the exponential graph (ie, the hockey stick). If you then zoom into the flat part of the graph, and choose a different y scale, you'll see a bend there. If you zoom out and choose a different y scale, you'll see your previous bend as part of the flat part. Unless there is a natural and obviously significant range of y values that, independent of the mathematical properties of the graph, is clearly what you actually care about, you can just make the "bend" in an exponential graph be anywhere you want it to be. Here is the graph of e^x from 0 to 10, showing the "bend" is at around x = 7. https://www.google.com/search?q=graph+of+e%5Ex+from+0+to+10&sca_esv=8ea0361af31360af&sca_upv=1&sxsrf=ADLYWILfr0kgB4okSkZ1j7gfeKB3Ti_hvA%3A1717657063276&ei=511hZrzMEPWuptQP9pGV-As&ved=0ahUKEwj82-iYs8aGAxV1l4kEHfZIBb8Q4dUDCBE&uact=5&oq=graph+of+e%5Ex+from+0+to+10&gs_lp=Egxnd3Mtd2l6LXNlcnAiGWdyYXBoIG9mIGVeeCBmcm9tIDAgdG8gMTAyBBAjGCcyCBAAGIAEGKIEMggQABiABBiiBDIIEAAYgAQYogQyCBAAGIAEGKIESP8JUMoGWMoGcAF4AZABAJgBiwGgAYsBqgEDMC4xuAEDyAEA-AEBmAICoAKVAcICChAAGLADGNYEGEeYAwCIBgGQBgiSBwMxLjGgB98E&sclient=gws-wiz-serp https://www.google.com/search?q=graph+of+e%5Ex+from+0+to+10&... And here is a graph of e^x from 0 to 100, showing the "bend" is at about 85: https://www.google.com/search?q=graph+of+e%5Ex+from+0+to+100&sca_esv=8ea0361af31360af&sca_upv=1&sxsrf=ADLYWIKWLsH_wO8fn-xLZG6hzjGUEvxlgA%3A1717657057015&ei=4V1hZqpOnfym1A_IuauhAg&ved=0ahUKEwjqwuqVs8aGAxUdvokEHcjcKiQQ4dUDCBE&uact=5&oq=graph+of+e%5Ex+from+0+to+100&gs_lp=Egxnd3Mtd2l6LXNlcnAiGmdyYXBoIG9mIGVeeCBmcm9tIDAgdG8gMTAwMgUQIRigATIFECEYoAEyBRAhGKABMgUQIRigATIFECEYoAEyBRAhGKsCMgUQIRirAjIFECEYqwJIgQhQgQZYgQZwAXgBkAEAmAGPAaABjwGqAQMwLjG4AQPIAQD4AQGYAgKgApkBwgIKEAAYsAMY1gQYR5gDAIgGAZAGCJIHAzEuMaAHvgc&sclient=gws-wiz-serp https://www.google.com/search?q=graph+of+e%5Ex+from+0+to+100...