A tangent is drawn on the curve of the function $y = x^2$ at the point $(x, y) = (3,9)$. The slope of the tangent is____________.
Tangent Slope Calculation for $y=x^2$
To find the slope of the tangent line to a curve at a specific point, we need to calculate the derivative of the function and evaluate it at the x-coordinate of that point.
Derivative of the Function
The given function is $y = x^2$. The derivative of $y$ with respect to $x$, denoted as $\frac{dy}{dx}$, gives the slope of the tangent line at any point $x$. Using the power rule for differentiation ($\frac{d}{dx}(x^n) = nx^{n-1}$): $ \frac{dy}{dx} = \frac{d}{dx}(x^2) = 2x $
Evaluating Slope at Point (3, 9)
The point given is $(3, 9)$. We need to find the slope at $x=3$. Substitute $x=3$ into the derivative: $ \text{Slope} = \frac{dy}{dx} \Big|_{x=3} = 2 \times 3 $ $ \text{Slope} = 6 $
Thus, the slope of the tangent to the curve $y = x^2$ at the point $(3, 9)$ is 6.
Important Questions from Functions Of Single Variable
- Let $f : R \to R$ be a twice-differentiable function and suppose its second derivative
satisfies $f''(x) > 0$ for all $x \in R$. Which of the following statements is/are ALWAYS
correct? - The gradient of $y = 3x^2 \sin(2x)$ at (0.2, 1) is __________ (rounded off to three decimal places).
- If $y = x^x$, then $\frac{dy}{dx}$ is
Given the function $$f(x) = |x| + |x - 1|,$$ For all the real values of x, which one of the following statements is CORRECT ?
- Given $x$ is real, identify all the even-functions among the following:
