Solve the problem.
-Find an equation of the tangent to the curve $f ( x ) = 2 x ^ { 2 } - 2 x + 1$ that has slope 2 .
A) $y = 2 x - 1$
B) $y = 2 x$
C) $y = 2 x + 2$
D) $y = 2 x + 1$

Question

Quizplus · Accepted Answer

To find the equation of the tangent line with a slope of 2, first, we need to find the derivative of $f(x)$ to determine the slope of the tangent at any point $x$. The derivative $f'(x) = 4x - 2$. Setting this equal to 2 (the given slope of the tangent) gives $4x - 2 = 2$. Solving for $x$ gives $x = 1$. Substituting $x = 1$ into the original function $f(x)$ gives $f(1) = 2(1)^2 - 2(1) + 1 = 1$. Therefore, the point of tangency is $(1, 1)$. Using the point-slope form of a line, $y - y_1 = m(x - x_1)$, where $m = 2$, $x_1 = 1$, and $y_1 = 1$, we get $y - 1 = 2(x - 1)$, which simplifies to $y = 2x - 1$.

Solve the Problem f(x)=2x2−2x+1f ( x ) = 2 x ^ { 2 } - 2 x + 1f(x)=2x2−2x+1

Solve the Problem $f ( x ) = 2 x ^ { 2 } - 2 x + 1$