Write the equation of the tangent line to the curve at the indicated point. As a check, graph both the function and the tangent line.
f(x) = x7 − 7 at x =
−1
7 x7
To find the equation of the tangent line to the curve at the indicated point, we need to first find the derivative of the function and then use it to find the slope of the tangent line.
Let's start by finding the derivative of the function f(x) = x^7 - 7.
To find the derivative, we can apply the power rule, which states that for any constant n, the derivative of x^n is n*x^(n-1).
Using the power rule, the derivative of x^7 is 7*x^(7-1) = 7*x^6.
Now we can substitute the value of x = -1 into the derivative to find the slope of the tangent line at that point.
Slope = 7*(-1)^6 = 7.
So, the slope of the tangent line is 7.
Next, to find the equation of the tangent line, we need to use the point-slope form of a line, which is given by y - y1 = m(x - x1), where (x1, y1) is the point on the curve (-1, f(-1)) and m is the slope of the tangent line.
Substituting the values, we have:
y - (-1) = 7(x - (-1))
y + 1 = 7(x + 1)
y + 1 = 7x + 7
y = 7x + 6.
So, the equation of the tangent line to the curve f(x) = x^7 - 7 at x = -1 is y = 7x + 6.
To graph both the function and the tangent line, plot the points that satisfy the equation of the function f(x) = x^7 - 7 and the tangent line y = 7x + 6 on the same coordinate system.