Linear approximation Find the linear approximation to the following functions at the given point a.


g(t) = √(2t + 9); a = -4

Verified step by step guidance

1. Identify the function and the point of approximation: The function is \( g(t) = \sqrt{2t + 9} \) and the point of approximation is \( a = -4 \).
2. Find the derivative of the function \( g(t) \). Use the chain rule to differentiate \( g(t) = \sqrt{2t + 9} \). The derivative \( g'(t) \) is \( \frac{d}{dt}(2t + 9)^{1/2} = \frac{1}{2}(2t + 9)^{-1/2} \cdot 2 \).
3. Evaluate the function and its derivative at the point \( a = -4 \). Calculate \( g(-4) = \sqrt{2(-4) + 9} \) and \( g'(-4) \) using the derivative found in the previous step.
4. Use the linear approximation formula \( L(t) = g(a) + g'(a)(t - a) \) to find the linear approximation. Substitute \( g(-4) \), \( g'(-4) \), and \( a = -4 \) into the formula.
5. Simplify the expression for \( L(t) \) to obtain the linear approximation of \( g(t) \) at \( t = -4 \). This will give you the equation of the tangent line that approximates \( g(t) \) near \( t = -4 \).

Key Concepts

Linear Approximation

Linear approximation is a method used to estimate the value of a function near a given point using the tangent line at that point. It is based on the idea that a function can be closely approximated by a linear function when the input is near a specific value. The formula for linear approximation is f(a) + f'(a)(x - a), where f'(a) is the derivative of the function at point a.

Derivative

The derivative of a function at a point measures the rate at which the function's value changes as its input changes. It is defined as the limit of the average rate of change of the function as the interval approaches zero. In the context of linear approximation, the derivative provides the slope of the tangent line, which is crucial for constructing the linear approximation.

Function Evaluation

Function evaluation involves calculating the output of a function for a specific input value. In the context of linear approximation, it is essential to evaluate both the function and its derivative at the point of interest (a) to find the necessary values for the linear approximation formula. For the function g(t) = √(2t + 9), evaluating g(-4) and g'(-4) will provide the required information for the approximation.