To find the slope of the curve given by the equation \( y = 2x^2 + 5x - 3 \) at the point \( (1, 4) \), we need to follow these steps: ### Step 1: Understand the Concept of Slope The slope of a curve at a given point is determined by the derivative of the function at that point. The derivative represents the rate of change of the function with respect to \( x \). In simpler terms, it tells us how steep the curve is at that specific point. ### Step 2: Differentiate the Function To find the slope, we first need to differentiate the function \( y \) with respect to \( x \). The function is: \[ y = 2x^2 + 5x - 3 \] Using the power rule of differentiation, which states that if \( y = ax^n \), then \( \frac{dy}{dx} = nax^{n-1} \), we differentiate each term: 1. The derivative of \( 2x^2 \) is \( 2 \cdot 2x^{2-1} = 4x \). 2. The derivative of \( 5x \) is \( 5 \cdot 1x^{1-1} = 5 \). 3. The derivative of the constant \( -3 \) is \( 0 \). Putting it all together, the derivative \( \frac{dy}{dx} \) is: \[ \frac{dy}{dx} = 4x + 5 \] ### Step 3: Evaluate the Derivative at the Given Point Now that we have the derivative, we need to evaluate it at \( x = 1 \) to find the slope at the point \( (1, 4) \): \[ \frac{dy}{dx} \bigg|_{x=1} = 4(1) + 5 = 4 + 5 = 9 \] ### Step 4: Conclusion The slope of the curve \( y = 2x^2 + 5x - 3 \) at the point \( (1, 4) \) is \( 9 \). Therefore, the correct option is: **D. 9** ### Explanation of Other Options - **A. 4**: This option is incorrect because it does not account for the contributions from both terms in the derivative. The slope is not simply the coefficient of \( x \) in the linear term. - **B. 6**: This option is also incorrect. It seems to be a miscalculation, possibly from incorrectly evaluating the derivative or misunderstanding the function's behavior. - **C. 7**: This option is incorrect as well. It may arise from an error in the arithmetic when adding the terms of the derivative. ### Common Pitfalls - **Misunderstanding the Derivative**: Students often confuse the derivative with the function itself. Remember, the derivative gives the slope, not the value of the function. - **Arithmetic Errors**: Simple addition or multiplication mistakes can lead to incorrect answers. Always double-check your calculations. - **Evaluating at the Wrong Point**: Ensure that you substitute the correct \( x \) value into the derivative. ### Revision Summary - The slope of a curve at a point is found using the derivative of the function. - Differentiate the function term by term using the power rule. - Evaluate the derivative at the specific \( x \) value to find the slope. - The correct slope for the curve \( y = 2x^2 + 5x - 3 \) at \( (1, 4) \) is \( 9 \).