Estimate the instantaneous rate of change at the point indicated.
y(t) = sqrt(3t+1)
t=1
y(t) = √(3+1)
y(t) = 2
To estimate the instantaneous rate of change at a specific point, we need to find the derivative of the function with respect to the independent variable, in this case, "t."
Let's find the derivative of the function y(t) = sqrt(3t+1) using the chain rule.
Step 1: Rewrite the given function using an exponent of 1/2 instead of square root notation:
y(t) = (3t + 1)^(1/2)
Step 2: Take the derivative of y(t) using the power rule. To differentiate (3t + 1)^(1/2), we apply the chain rule. The chain rule states that if we have a function of the form (f(g(t))), the derivative is given by f'(g(t)) * g'(t), where f'(x) and g'(t) are the derivatives of f(x) and g(t), respectively.
Let's apply the chain rule:
dy/dt = (1/2)(3t + 1)^(-1/2) * d/dt(3t + 1)
Step 3: Differentiate the term (3t + 1) with respect tot. The derivative of 3t with respect to t is 3, and the derivative of 1 with respect to t is 0 (constant term). Therefore, the derivative of (3t + 1) is simply 3.
dy/dt = (1/2)(3t + 1)^(-1/2) * 3
Step 4: Simplify the expression:
dy/dt = (3/2)(3t + 1)^(-1/2)
Now that we have the derivative expression, we can estimate the instantaneous rate of change at the given point (t = 1).
Substitute t = 1 into the derivative expression:
dy/dt = (3/2)(3(1) + 1)^(-1/2)
= (3/2)(4)^(-1/2)
= (3/2)*(1/2)
= 3/4
Therefore, the estimated instantaneous rate of change at t = 1 is 3/4.