Final answer:
To find the derivative of the function f(x) = 3x² + 1, use the definition of the derivative by taking the limit as h approaches 0. The derivative is 6x. To find the equation of the tangent line to the curve y = f(x) at (1, 2), use the point-slope form of a linear equation. The equation of the tangent line is y = 6x - 4.
Step-by-step explanation:
To find the derivative of the function f(x) = 3x² + 1 using the definition of the derivative, we can use the following formula:
f'(x) = lim(h→0) [f(x + h) - f(x)] / h
Substitute the given function f(x) with 3x² + 1:
f'(x) = lim(h→0) [(3(x + h)² + 1) - (3x² + 1)] / h
Expand and simplify the expression:
f'(x) = lim(h→0) [(3x² + 6hx + 3h² + 1) - (3x² + 1)] / h
Combine like terms:
f'(x) = lim(h→0) [6hx + 3h²] / h
Cancel out the h in the numerator and denominator:
f'(x) = lim(h→0) [6x + 3h]
Take the limit as h approaches 0:
f'(x) = 6x
Therefore, the derivative of f(x) = 3x² + 1 is f'(x) = 6x.
To find the equation of the tangent line to the curve y = f(x) at (1, 2), we will use the point-slope form of a linear equation. We know that the slope of the tangent line is equal to the derivative of f(x) at x = 1, which is 6. Now we have the slope (m = 6) and a point (1, 2) on the line, so we can substitute these values into the point-slope form equation:
y - y₁ = m(x - x₁)
Substitute y₁ = 2, x₁ = 1, and m = 6:
y - 2 = 6(x - 1)
Simplify the equation:
y - 2 = 6x - 6
Add 2 to both sides of the equation:
y = 6x - 4
Therefore, the equation of the tangent line to the curve y = f(x) at (1, 2) is y = 6x - 4.