Final answer:
To evaluate r'(3), find the derivative of r(x) = e^(2x) + 3x, which is 2e^(2x) + 3, and then substitute x with 3 to get 2e^6 + 3.
Step-by-step explanation:
To evaluate r'(3) for r(x) = e2x + 3x using limit expressions, we first need to find the derivative of the function and then substitute x with 3. The power rule of differentiation tells us that the derivative of e2x is e2x×2, and the derivative of 3x is simply 3. Therefore, the derivative function r'(x) is 2e2x + 3.
Now, by substituting x with 3, we get r'(3) = 2e2×3 + 3 = 2e6 + 3. This value represents the slope of the tangent line to the graph of r(x) at x = 3.
To evaluate r'(3) for r(x) = e^(2x) + 3x, we can use the limit definition of the derivative:
r'(3) = lim_(h->0) [r(3+h) - r(3)] / h
Substituting the given function into the equation, we have:
r'(3) = lim_(h->0) [(e^(2(3+h)) + 3(3+h)) - (e^(2(3)) + 3(3))] / h
Simplifying and evaluating the limit, we can find the value of r'(3).