Use the definition of the instantaneous rate of change to the expression for the slope of the tangent line to find the function 2 square root x plus 3 at any point (x,f(x)). Then proceed to find the equation of the tangent line at x=4. graph the function and the tangent line you found on the same coordinate system. Does your answer makes sense?

Here is the graph of the function \( f(x) = 2\sqrt{x} + 3 \) along with the tangent line at \( x = 4 \). The blue curve represents the function, and the red dashed line represents the tangent line with the equation \( y = \frac{1}{2}x + 5 \). The green point indicates the point of tangency at \( (4, 7) \).

### Does the Answer Make Sense?
Yes, the answer makes sense. The tangent line touches the curve exactly at the point \( (4, 7) \) and has the correct slope of \( \frac{1}{2} \), as calculated earlier. The tangent line approximates the behavior of the function near \( x = 4 \), which aligns with the definition of the derivative as the instantaneous rate of change.

Would you like more details on any step, or do you have any questions?

Here are some related questions:

1. How would you find the equation of the tangent line at a different point, say \( x = 9 \)?
2. What is the significance of the slope \( \frac{1}{2} \) in the context of the function at \( x = 4 \)?
3. How does the tangent line approximation behave as you move further away from the point of tangency?
4. What is the geometric interpretation of the derivative at a point in relation to the curve?
5. How can the derivative be used to determine concavity and inflection points of a function?

**Tip:** When analyzing the behavior of a function, graphing both the function and its tangent lines at different points can provide valuable insights into the nature of the function and its rates of change.

Explore how to find the equation of a tangent line to the function f(x) = 2sqrt(x) + 3 at any point (x, f(x)), specifically at x = 4. Graph the function and tangent line on the same coordinate system, understanding the concept of derivatives and tangent lines in calculus.

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