Question

If y = 5x\(^2\) + 4, then at the point with x-coordinate 2, the slope is

• A. \(3/2\sqrt{14}\)
• B. \(1/2\sqrt{14}\)
• C. 20
• D. 1

Hint:

In competitive exams, be aware of potential typos in the questions. If a given point (x, y) does not satisfy the equation of the curve, double-check if you only need the x-coordinate for the calculation (like finding a slope). Proceed with the given x-value and see if the result matches an option.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
The slope of the tangent line to a curve at a specific point is given by the value of the first derivative of the function at that point.
The question provides a function y = 5x\(^2\) + 4 and asks for the slope at the point where x = 2. The y-coordinate mentioned in the image is likely a typo, as it does not lie on the curve, but it is not needed to find the slope.
Step 2: Key Formula or Approach:
The slope 'm' of the tangent to the curve y = f(x) at x = a is given by: \[ m = f'(a) = \left. \frac{dy}{dx} \right|_{x=a} \] We will use the power rule for differentiation: \(\frac{d}{dx}(x^n) = nx^{n-1}\).
Step 3: Detailed Explanation:
The given function is: \[ y = 5x^2 + 4 \] First, we find the derivative of y with respect to x, which represents the slope function: \[ \frac{dy}{dx} = \frac{d}{dx}(5x^2 + 4) \] \[ \frac{dy}{dx} = 5 \cdot (2x^{2-1}) + 0 \] \[ \frac{dy}{dx} = 10x \] Now, we need to find the slope at the point where the x-coordinate is 2. We substitute x = 2 into the derivative: \[ \text{Slope} (m) = 10(2) \] \[ m = 20 \] The y-coordinate of the point on the curve would be y = 5(2)\(^2\) + 4 = 24. The point is (2, 24). The slope at this point is 20.
Step 4: Final Answer:
The slope of the curve at x = 2 is 20. So, option (C) is correct.