Question

hw 9 - the tangent line section 2.4: problem 3 (1 point)
let $f(x)=5x^{3}-x^{2}-7x + 7$.(a) find $f(x)$.$f(x)=square$(b) find the slope of the line tangent to the graph of $f$ at $x = 6$.slope at $x = 6:square$ (use at least 2 decimal places)(c) find an equation of the line tangent to the graph of $f$ at $x = 6$.tangent line: $y=square$

Explanation:

Step1: Differentiate \(f(x)\)

Using the power - rule \((x^n)^\prime=nx^{n - 1}\), for \(f(x)=5x^{3}-x^{2}-7x + 7\), we have \(f^\prime(x)=5\times3x^{2}-2x-7=15x^{2}-2x - 7\).

Step2: Find the slope at \(x = 6\)

Substitute \(x = 6\) into \(f^\prime(x)\). \(f^\prime(6)=15\times6^{2}-2\times6 - 7=15\times36-12 - 7=540-12 - 7 = 521\).

Step3: Find the point on the function at \(x = 6\)

Find \(f(6)\): \(f(6)=5\times6^{3}-6^{2}-7\times6 + 7=5\times216-36 - 42 + 7=1080-36 - 42 + 7=1009\).

Step4: Find the equation of the tangent line

Use the point - slope form \(y - y_1=m(x - x_1)\), where \(m = 521\), \(x_1 = 6\), and \(y_1 = 1009\).
\(y-1009=521(x - 6)\)
\(y-1009=521x-3126\)
\(y=521x-3126 + 1009\)
\(y=521x-2117\)

Answer:

(a) \(15x^{2}-2x - 7\)
(b) \(521.00\)
(c) \(521x-2117\)