The value of k is 3.5 for which y = 3x + k is a tangent to the curve y = 2x^2 - 3x + 8.
The derivative helps determine the slope of the tangent at any point on the curve.
The slope of the tangent line y = 3x + k is 3.
dy/dx = 4x - 3
3 = 4x - 3
6 = 4x
x = 1.5
Plug this into the curve's original equation.
[tex]y = 2x^2 - 3x + 8\\y = 2(1.5)^2 - 3(1.5) + 8\\y = 8[/tex]
Therefore, the tangent line y = 3x + k and the curve y = 2x^2 - 3x + 8 intersect at the point (1.5, 8). This is the point of tangency.
[tex]y = 3x + k\\\\8 = 3(1.5) + k\\\\k = 8- 4.5\\\\k = 3.5[/tex]
Thus, the value of k is 3.5 for which y = 3x + k is a tangent to the curve y = 2x^2 - 3x + 8.
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