Question
Will 𝑦 =5 (3/5)^x be steeper than, less steep than or the same steepness as the basic curve of 𝑦 = (3/5)^x
Answers
Answered by
GPT 3.5
To determine the steepness of the curve 𝑦 = 5 (3/5)^x compared to the basic curve 𝑦 = (3/5)^x, we can look at the derivative of each function.
The derivative of the basic curve 𝑦 = (3/5)^x is given by:
dy/dx = ln(3/5) * (3/5)^x
The derivative of the modified curve 𝑦 = 5 (3/5)^x is given by:
dy/dx = 5 * (ln(3/5)) * (3/5)^x
Comparing the two derivatives, we see that the modified curve has a steeper slope than the basic curve. This is because the slope of the modified curve is scaled by a factor of 5, which results in a faster rate of increase as x changes.
Therefore, the curve 𝑦 = 5 (3/5)^x will be steeper than the basic curve 𝑦 = (3/5)^x.
The derivative of the basic curve 𝑦 = (3/5)^x is given by:
dy/dx = ln(3/5) * (3/5)^x
The derivative of the modified curve 𝑦 = 5 (3/5)^x is given by:
dy/dx = 5 * (ln(3/5)) * (3/5)^x
Comparing the two derivatives, we see that the modified curve has a steeper slope than the basic curve. This is because the slope of the modified curve is scaled by a factor of 5, which results in a faster rate of increase as x changes.
Therefore, the curve 𝑦 = 5 (3/5)^x will be steeper than the basic curve 𝑦 = (3/5)^x.
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