Using the Limit Definition to Find the Tangent Line at a Given Point

To identify the slope of the tangent line to a curve at a specified point, one can resort to the limit definition. The method involves computing the limit of the difference quotient as the interval shrinks closer to zero. This allows for the calculation of the instantaneous rate of change, reflecting the slope of the tangent line.

The problems about Using the Limit Definition to Find the Tangent Line at a Given Point

Topic	Problem	Solution
None	Use $Δ y \approx f^{'} (x) Δ x$ to …	The problem is asking for a decimal approximation of the cube root of 8.66 using the linear approxi…
None	'The function $f (x)$ changes value when $x$ chang…	First, we need to find the value of the function $f (x)$ at $x_{0}$ and $x_{0} + d x$ . Given $x_{0} = 1$ …
None	(a) Find the slope of the curve $y = x^{2} - 3 x - 4$ a…	To find the slope of the curve at a given point, we need to find the derivative of the function at …
None	Use $Δ y \approx f^{'} (x) Δ x$ to …	We are given the function $f (x) = \sqrt{x}$ and we want to find an approximation for $f (107)$ .
None	For $y = f (x) = 8 x^{3}, x = 4$ , and $Δ x = 0.02$ fi…	Given the function $y = f (x) = 8 x^{3}$ , where $x = 4$ , and $Δ x = 0.02$ .
None	For $y = f (x) = 6 x^{3}, x = 2$ , and $Δ x = 0.06$ fi…	Given the function $y = f (x) = 6 x^{3}$ , the value of $x = 2$ , and $Δ x = 0.06$ .