for the function $f(x)=10x^{2}-3x - 4$, find the equation of the tangent line at $x = 9$.
for the function $f(x)=10x^{2}-3x - 4$, find the equation of the tangent line at $x = 9$.
Answer
Explanation:
Step1: Find the derivative of the function
The derivative of (f(x)=10x^{2}-3x - 4) using the power rule ((x^{n})^\prime=nx^{n - 1}) is (f^\prime(x)=20x-3).
Step2: Find the slope of the tangent line at (x = 9)
Substitute (x = 9) into (f^\prime(x)): (m=f^\prime(9)=20\times9-3=180 - 3=177).
Step3: Find the (y) - coordinate of the point on the function at (x = 9)
Substitute (x = 9) into (f(x)): (y=f(9)=10\times9^{2}-3\times9-4=10\times81-27 - 4=810-27-4 = 779).
Step4: Use the point - slope form (y - y_1=m(x - x_1)) to find the tangent line equation
Here (x_1 = 9,y_1 = 779,m = 177). The point - slope form is (y-779=177(x - 9)). Expand it: (y-779=177x-1593). Then (y=177x-1593 + 779).
Answer:
(y = 177x-814)