suppose that $f$ is a function defined as $f(x)=-3x^{3}+x^{2}+7$. find an equation for the tangent line…

suppose that $f$ is a function defined as $f(x)=-3x^{3}+x^{2}+7$. find an equation for the tangent line drawn to the graph of $f$ at $x = 1$. use exact values.

suppose that $f$ is a function defined as $f(x)=-3x^{3}+x^{2}+7$. find an equation for the tangent line drawn to the graph of $f$ at $x = 1$. use exact values.

Answer

Explanation:

Step1: Find the derivative of the function

The derivative of (f(x)=-3x^{3}+x^{2}+7) is (f^{\prime}(x)=-9x^{2}+2x) (using the power rule ((x^{n})^\prime = nx^{n - 1})).

Step2: Evaluate the function and its derivative at (x = 1)

  • Evaluate (f(1)): (f(1)=-3(1)^{3}+(1)^{2}+7=-3 + 1+7=5)
  • Evaluate (f^{\prime}(1)): (f^{\prime}(1)=-9(1)^{2}+2(1)=-9 + 2=-7)

Step3: Use the point - slope form of a line

The point - slope form is (y - y_{0}=m(x - x_{0})), where ((x_{0},y_{0})=(1,5)) and (m = f^{\prime}(1)=-7). Substituting into the formula: (y - 5=-7(x - 1)) Expand: (y-5=-7x + 7)

Step4: Write the equation in slope - intercept form

Add 5 to both sides: (y=-7x+7 + 5)

Answer:

(y=-7x + 12)