Questions: Find an equation of the tangent line to the given function at (a=1). (f(x)=x^2-7) (y=)

Solution Steps

To find the equation of the tangent line, we first need the derivative of \(f(x)\), which gives the slope of the tangent line at any point \(x\). The function is: \[ f(x) = x^{2} - 7 \] The derivative \(f'(x)\) is: \[ f'(x) = \frac{d}{dx}(x^{2} - 7) = 2x \]

The slope of the tangent line at \(a = 1\) is given by \(f'(1)\): \[ f'(1) = 2(1) = 2 \] So, the slope \(m\) of the tangent line is \(2\).

To find the point on the curve at \(a = 1\), evaluate \(f(1)\): \[ f(1) = (1)^{2} - 7 = 1 - 7 = -6 \] So, the point is \((1, -6)\).

Using the point-slope form of a line, \(y - y_1 = m(x - x_1)\), where \(m\) is the slope and \((x_1, y_1)\) is the point: \[ y - (-6) = 2(x - 1) \] Simplify the equation: \[ y + 6 = 2x - 2 \] \[ y = 2x - 8 \]

Final Answer