Question

for ( f(x) = x^4 ), the instantaneous rate of change is known to be 4 at ( x = 1 ). find the equation of the tangent line to the graph of ( y = f(x) ) at the point with x-coordinate 1. the equation of the tangent line to the graph at the point with x-coordinate 1 is (square). (type an equation. use integers or fractions for any numbers in the equation.)

Explanation:

Step1: Find the y - coordinate

We know the function \(f(x)=x^{4}\). To find the \(y\) - coordinate of the point on the graph when \(x = 1\), we substitute \(x=1\) into the function.
\(f(1)=1^{4}=1\). So the point of tangency is \((1,1)\).

Step2: Use the point - slope form of a line

The point - slope form of a line is \(y - y_{1}=m(x - x_{1})\), where \((x_{1},y_{1})\) is a point on the line and \(m\) is the slope of the line.
We know that the instantaneous rate of change of the function at \(x = 1\) is the slope of the tangent line at \(x = 1\). So \(m = 4\), and the point \((x_{1},y_{1})=(1,1)\).
Substitute these values into the point - slope form:
\(y-1 = 4(x - 1)\)

Step3: Simplify the equation

Expand the right - hand side: \(y-1=4x-4\).
Then add 1 to both sides of the equation: \(y=4x-4 + 1\), which simplifies to \(y = 4x-3\).

Answer:

\(y = 4x-3\)