Answer :
The slope of the tangent line to the curve is 2 when the x-value of the function is x = 1
How can the slope of the tangent line to a curve be found?
The slope of the tangent line to a curve at a given point is equal to the derivative of the function at that point.
Given the points on the graph, it appears that the function could be a quadratic function of the form f(x) = a·x² + b·x + c. The derivative of this function would be f'(x) = 2·a·x + b
To find the value of x where the slope of the tangent line is equal to 2, we would need to solve the equation 2·a·x + b = 2 for x
The points on the graph, indicates that we get;
f(0) = 5.5, therefore; a × 0² + b × 0 + c = -5.5
c = -5.5
f(1) = -3, therefore; a × 1² + b × 1 - 5.5 = -3
a + b - 5.5 = -3
a + b = 2.5
f(3) = -1, therefore; a × 3² + b × 3 - 5.5 = -1
9·a + 3·b - 5.5 = -1
9·a + 3·b = 4.5
Solving, we get; a = -0.5, b = 3
The quadratic equation is f(x) = -0.5·x² + 3·x - 5.5
f'(x) = 2 × -0.5·x + 3
f'(x) = -x + 3
When the slope is 2, we get;
f'(x) = 2, therefore; 2 = -x + 3
x = 3 - 2 = 1
The slope of the tangent line to the curve is 2 when x = 1
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