Question

How do you find the slope of the tangent line to the graph of the function $h\left( t \right) = {t^2} + 3$ at $\left( { - 2,7} \right)$?

Answer 1

Here we need to find the value of the slope of the graph of the given function. We will first differentiate the given function with respect to the variable used in the function and then we will substitute the value of the coordinates of the given point at which the tangent was drawn. After differentiating the function, we will further simplify it using mathematical operations like addition, subtraction, multiplication, and division. After simplifying the terms, we will get the required value of the slope.

Complete step by step solution:
Here we need to find the value of the slope of the graph of the given function and the given function is $h\left( t \right) = {t^2} + 3$.
Now, we will first differentiate the given function with respect to $t$.
$\Rightarrow \dfrac{{dh\left( t \right)}}{{dt}} = \dfrac{{d\left( {{t^2} + 3} \right)}}{{dt}}$
Now, we will differentiate each term using the formula $\dfrac{d}{{dx}}\left( {{x^n}} \right) = n{x^{n - 1}}$ and $\dfrac{d}{{dx}}\left( k \right) = 0$. Therefore, we get
$\Rightarrow \dfrac{{dh\left( t \right)}}{{dt}} = 2t$
Here, we need to find the value of the slope of the tangent at the point $\left( { - 2,7} \right)$. So we will substitute the value of the coordinates of this point in the derivative of the function.
$\Rightarrow {\left. {\dfrac{{dh\left( t \right)}}{{dt}}} \right|_{\left( { - 2,7} \right)}} = 2 \times - 2$
On multiplying the numbers, we get
$\Rightarrow {\left. {\dfrac{{dh\left( t \right)}}{{dt}}} \right|_{\left( { - 2,7} \right)}} = - 4$

Hence, the value of the slope of the tangent is equal to -4.

Note:
Here we have obtained the value of the slope of the tangent of the graph of the given function. Here slope of the line is defined as the measurement of the inclination of the line with the coordinate axis and it also measures the steepness of the line. The slope of the tangent measured the inclination of the tangent with the coordinate axis. We know that the derivative of a given function is defined as the change of the given function with respect to the variable used in the given function. Differentiation is defined as the action of calculating the derivative of a function.