Ascertaining a tangent line to a curve essentially requires pinpointing a point on the curve and employing the mathematical study of change, known as calculus, in particular differentiation, to determine the incline of the line at that specific point. This gradient, coupled with the coordinates of the point, can then be replaced into the slope-point form of the line equation.
| Topic | Problem | Solution |
|---|---|---|
| None | Use implicit differentiation to find $y^{\prime}$… | Differentiate the given equation implicitly with respect to x. This involves applying the product r… |
| None | Use implicit differentiation to find $d y / d x$.… | Use implicit differentiation to find \(\frac{d y}{d x}\). |
| None | Find an equation of the line tangent to the graph… | First, we need to find the derivative of the function \(f(x) = 2x^3\). The derivative of a function… |
| None | Assuming that the equations define $x$ and $y$ im… | Given the equations \(x=t^{5}+t\) and \(y+4 t^{5}=4 x+t^{3}\) at \(t=2\), we are asked to find the … |
| None | Find the equation of the tangent line at the give… | First, we need to find the derivative of the given function. The derivative of \(3y^2 - \sqrt{x} = … |
| None | Find an equation of the tangent line to the given… | Given the function \(y=\frac{e^{8 x}}{x}\) and the point \(\left(\frac{1}{8}, 8 e\right)\), we are … |
| None | Find the linearization $L(x, y)$ of the function … | Given the function \(f(x, y) = x^2 + y^2 + 1\), we want to find its linearization at the point (4,1… |
| None | Find the equation of the line tangent to the grap… | Let's find the derivative of the function \(f(x)=(\ln x)^{4}\). |
| None | possible The given point is on the curve. Find th… | We are given the curve \(6 x^{2}+7 x y+2 y^{2}+13 y-6=0\) and the point \((-1,0)\). We are asked to… |
| None | Find an equation for the tangent to the curve at … | Given the function \(f(x) = 10 \sqrt{x} - x + 7\) and the point \((100,7)\). |
| None | Find an equation for the tangent plane to the sur… | Given the surface equation \(z+5=xy^3\cos(z)\), we want to find the equation of the tangent plane a… |
| None | Find a linearization that will replace the functi… | We are given the function \(f(x) = \sqrt[4]{x}\) and we are asked to find a linearization of this f… |
| None | Let $f(x)=6 x^{2}$ a) Find the linearization $L(x… | The function given is \(f(x) = 6x^{2}\). |
| None | Find the slope of the tangent line to the curve \… | First, we need to find the derivative of the given equation. The derivative of a function gives us … |
| None | Assuming that the equation defines $x$ and $y$ im… | Given the equations \(x^{3}+4 t^{2}=65\) and \(2 y^{3}-3 t^{2}=80\), we are asked to find the slope… |
| None | Find (a) the slope of the curve at the given poin… | Given the function \(y = -2 - 4x^2\) and the point \(P(3,-38)\). |
| None | a. Find an equation of the tangent line at $x=a$.… | We are given the function \(y=e^{x}\) and \(a=\ln 10\). |
| None | Find an equation of the tangent line to the graph… | Find the derivative of the function using the chain rule: \(\frac{dy}{dx} = -2xe^{-x^2}\) |
| None | The line with equation $y=-2 x+c$ is tangent to t… | Rearrange the equation of the line: $y = -2x + c$ to get $-2x = y - c$ |
| None | The function $f$ is given by $f(x)=\frac{1}{3} e^… | \(f(x) = \frac{1}{3} e^{-3x} + 1\) |
| None | For the function $f(x)=x^{2}+2$, find the equatio… | First, find the slope of the tangent line by taking the derivative of the function: \(f'(x) = 2x\) |
| None | 15 Point $A$ lies on the curve $y=x^{2}+5 x+8$ Th… | Find the coordinates of point A: (-4, 4) |
| None | Find the equation of the tangent line to the func… | Find the derivative of the function: \(f'(x) = -18x + 3\) |
| None | Fid the value of \( k \) such that the tangent li… | \( f'(x) = \frac{-2(k+x)}{x^{3}} \) |
| None | Find the equation of the tangent line to \( g(x)=… | \(g'(x)=3x^{2}\) |
| None | Find the equation of the tangent line to \( g(x)=… | Find \( g'(x) \) by taking the derivative of \( g(x)=x^{2} \). |
| None | 10. Calculate the tangent planes of the following… | \[ \frac{\partial f}{\partial x} = 2x, \frac{\partial f}{\partial y} = 2y \Rightarrow \frac{\partia… |