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.
| Topic | Problem | Solution |
|---|---|---|
| None | Use $\Delta y \approx f^{\prime}(x) \Delta x$ 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 $x_{0}+dx$. 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 $\Delta y \approx f^{\prime}(x) \Delta x$ 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 $\Delta x=0.02$ fi… | Given the function \(y=f(x)=8 x^{3}\), where \(x=4\), and \(\Delta x=0.02\). |
| None | For $y=f(x)=6 x^{3}, x=2$, and $\Delta x=0.06$ fi… | Given the function \(y = f(x) = 6x^3\), the value of \(x = 2\), and \(\Delta x = 0.06\). |