f(t) = t^2 − 1
Find the equation of the line tangent to the graph of f(t) at t = 5.
Enter the equation of the tangent line here (in terms of the variable t):
y =
To find the equation of the line tangent to the graph of f(t) at t = 5, we need to find the slope of the tangent line at that point and use the point-slope form of a linear equation.
Step 1: Find the derivative of f(t):
f'(t) = 2t
Step 2: Evaluate the derivative at t = 5:
f'(5) = 2(5) = 10
Step 3: Use the point-slope form of a linear equation with the point (5, f(5)):
y - f(5) = m(t - 5)
Step 4: Substitute the values of f(5) = 5^2 - 1 = 24 and m = 10:
y - 24 = 10(t - 5)
Simplifying the equation, we get:
y - 24 = 10t - 50
Finally, rearrange the equation to get the equation of the tangent line:
y = 10t - 26
Therefore, the equation of the tangent line to the graph of f(t) at t = 5 is:
y = 10t - 26.