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
- The gradient of $y = 3x^2 \sin(2x)$ at (0.2, 1) is __________ (rounded off to three decimal places).
- Let $ f(x) = x - [x] $, where $ x \ge 0 $ and $ [x] $ is the greatest integer not larger than x. Then $ f(x) $ is a
Consider the hyperbolic functions in Group – 1 and their definitions in Group - 2.
Group - 1 Group - 2 P $\tanh x$ I $\frac{e^x + e^{-x}}{e^x - e^{-x}}$ Q $\coth x$ II $\frac{2}{e^x + e^{-x}}$ R $\text{sech } x$ III $\frac{2}{e^x - e^{-x}}$ S $\text{cosech } x$ IV $\frac{e^x - e^{-x}}{e^x + e^{-x}}$ The correct combination is
- The equation of the straight line representing the tangent to the curve $y = x^2$ at the point $(1,1)$ is
- The figure which represents $y = \frac{\sin x}{x}$ for $x > 0$ (x in radians) is
