Which of the following functions has a rate of change that stays the same?
A)y = x^2 – 5x –14
B)y = 19x – 10
C)y = 5^x
D)y = 0.03x^2 + 11x + 1
Answer is B
a constant rate (slope) means a linear function. So, which of those is linear?
The rate of change of a function can be determined by finding its derivative. The derivative represents the slope of the function at any given point.
Let's find the derivatives of the given functions to determine which one has a rate of change that stays the same:
A) For the function y = x^2 - 5x - 14, taking the derivative will give us:
dy/dx = 2x - 5
The derivative is linear, so the rate of change is not constant.
B) For the function y = 19x - 10, the derivative is:
dy/dx = 19
The derivative is constant, indicating that the rate of change stays the same.
C) For the function y = 5^x, the derivative is:
dy/dx = ln(5) * 5^x
The derivative is exponentially increasing, so the rate of change is not constant.
D) For the function y = 0.03x^2 + 11x + 1, the derivative is:
dy/dx = 0.06x + 11
The derivative is linear, so the rate of change is not constant.
Therefore, the function with a rate of change that stays the same is B) y = 19x - 10.
To determine which function has a rate of change that stays the same, we need to find the slope (rate of change) of each function. The slope is a measure of how much the function changes for each unit change in the variable.
Let's analyze each function:
A) y = x^2 – 5x – 14
To find the slope, we take the derivative of the function:
y' = 2x - 5
Since the derivative is not a constant, the rate of change is not the same throughout the function.
B) y = 19x – 10
The equation is in the form y = mx + b, where m is the slope. In this case, m = 19, which is a constant. So, this equation has a rate of change that stays the same.
C) y = 5^x
To find the slope, we take the derivative using logarithmic differentiation:
ln(y) = ln(5^x)
ln(y) = x * ln(5)
Differentiating both sides:
(1/y) * dy/dx = ln(5)
dy/dx = y * ln(5)
Since the derivative depends on the value of y, the rate of change is not constant.
D) y = 0.03x^2 + 11x + 1
To find the slope, we take the derivative of the function:
y' = 0.06x + 11
Since the derivative is not a constant, the rate of change is not the same throughout the function.
Therefore, the function with a rate of change that stays the same is B) y = 19x - 10.