Tangent Line Properties at a Point
Finding the Slope of the Tangent
To determine the properties of the tangent line to the curve defined by the equation $y = 5x^2 + 3$ at the point $(0, 3)$, we first need to find the slope of the tangent at this specific point.
The slope of the tangent line at any point on a curve is given by the derivative of the function, $\frac{dy}{dx}$.
First, find the derivative of $y = 5x^2 + 3$ with respect to $x$:
$\frac{dy}{dx} = \frac{d}{dx}(5x^2 + 3)$
$\frac{dy}{dx} = 10x$
Evaluating the Slope at x = 0
Now, substitute the x-coordinate of the given point, $x = 0$, into the derivative to find the slope ($m$) at that point:
$m = \frac{dy}{dx}\Big|_{x=0} = 10(0)$
$m = 0$
Interpreting the Slope
A slope of $0$ indicates that the tangent line is horizontal.
A horizontal line is parallel to the x-axis.
Analyzing the Options
Based on the calculated slope:
- Option 1: The tangent equation is $y - 3 = 0(x - 0)$, which is $y = 3$. This line does not pass through $(0, 0)$.
- Option 2: The slope is 0, not +1.
- Option 3: The slope is 0, meaning the tangent line is parallel to the x-axis. This is consistent with our findings.
- Option 4: The slope is 0, not -1.
Therefore, the tangent line at the point $(0, 3)$ is parallel to the x-axis.
Important Questions from Calculus
The ratio of volume to surface area of solid semi sphere is related to its radius through
The value of \(\int^2_0\int^x_0y\ dy\ dx\)
Find the slope of normal to the curve y = x2 + 7x at (1, 8).
Find the equation of normal to the curve y = 4x - 3x2 at (2, -4).
- \(\mathop {\lim }\limits_{\theta \to \frac{\pi }{2}} \frac{{\log \left( {\theta - \frac{\pi }{2}} \right)}}{{\tan \theta }}\)
