Find tangent at given point (polynomial/algebraic)

A curve has equation \(y = \frac{4}{3x - 4}\) and \(P(2, 2)\) is a point on the curve.

1. Find the equation of the tangent to the curve at \(P\). [4]
2. Find the angle that this tangent makes with the \(x\)-axis. [2]

The curve \(C\) has equation \(y = 2e^x + 3x^2 + 2\). The point \(A\) with coordinates \((0, 4)\) lies on \(C\). Find the equation of the tangent to \(C\) at \(A\). [5]

\includegraphics{figure_1} Figure 1 shows the curve with equation \(y = 8x - x^{\frac{3}{2}}\), \(x \geq 0\). The curve meets the \(x\)-axis at the origin, \(O\), and at the point \(A\).

1. Find the \(x\)-coordinate of \(A\). [3]
2. Find the gradient of the tangent to the curve at \(A\). [4]

The curve with equation \(y = 2x^3 - 8x^{\frac{1}{3}}\) has a minimum at the point \(A\).

1. Find \(\frac{dy}{dx}\). [3]
2. Find the \(x\)-coordinate of \(A\). [3]

The point \(B\) on the curve has \(x\)-coordinate 2.

1. Find an equation for the tangent to the curve at \(B\) in the form \(y = mx + c\). [6]

\includegraphics{figure_3} Figure 3 shows part of the curve \(C\) with equation $$y = \frac{3}{5}x^2 - \frac{1}{4}x^3.$$ The curve \(C\) touches the \(x\)-axis at the origin and passes through the point \(A(p, 0)\).

1. Show that \(p = 6\). [1]
2. Find an equation of the tangent to \(C\) at \(A\). [4]

The curve \(C\) has a maximum at the point \(P\).

1. Find the \(x\)-coordinate of \(P\). [2]

The shaded region \(R\), in Fig. 3, is bounded by \(C\) and the \(x\)-axis.

1. Find the area of \(R\). [4]

Find the equation of the tangent to the curve \(y = 6\sqrt{x}\) at the point where \(x = 16\). [5]

The gradient of a curve is given by \(\frac{dy}{dx} = 4x + 3\). The curve passes through the point \((2, 9)\).

1. Find the equation of the tangent to the curve at the point \((2, 9)\). [3]
2. Find the equation of the curve and the coordinates of its points of intersection with the \(x\)-axis. Find also the coordinates of the minimum point of this curve. [7]
3. Find the equation of the curve after it has been stretched parallel to the \(x\)-axis with scale factor \(\frac{1}{2}\). Write down the coordinates of the minimum point of the transformed curve. [3]

\includegraphics{figure_1} Fig. 9 shows a sketch of the graph of \(y = x^3 - 10x^2 + 12x + 72\).

1. Write down \(\frac{dy}{dx}\). [2]
2. Find the equation of the tangent to the curve at the point on the curve where \(x = 2\). [4]
3. Show that the curve crosses the \(x\)-axis at \(x = -2\). Show also that the curve touches the \(x\)-axis at \(x = 6\). [3]
4. Find the area of the finite region bounded by the curve and the \(x\)-axis, shown shaded in Fig. 9. [4]

Find the equation of the tangent to the curve \(y = 6\sqrt{x}\) at the point where \(x = 16\). [5]

\includegraphics{figure_2} Figure 2 shows part of the curve \(C\) with equation \(y = \text{f}(x)\), where $$\text{f}(x) = 0.5e^x - x^2.$$ The curve \(C\) cuts the \(y\)-axis at \(A\) and there is a minimum at the point \(B\).

1. Find an equation of the tangent to \(C\) at \(A\). [4]

The \(x\)-coordinate of \(B\) is approximately 2.15. A more exact estimate is to be made of this coordinate using iterations \(x_{n+1} = \ln g(x_n)\).

1. Show that a possible form for \(g(x)\) is \(g(x) = 4x\). [3]
2. Using \(x_{n+1} = \ln 4x_n\), with \(x_0 = 2.15\), calculate \(x_1\), \(x_2\) and \(x_3\). Give the value of \(x_3\) to 4 decimal places. [2]

The curve \(C\) has equation \(y = 2e^x + 3x^2 + 2\). The point \(A\) with coordinates \((0, 4)\) lies on \(C\). Find the equation of the tangent to \(C\) at \(A\). [5]

f(x) = \(x + \frac{e^x}{5}\), \(x \in \mathbb{R}\).

1. Find f'(x). [2]

The curve \(C\), with equation \(y = \)f(x), crosses the \(y\)-axis at the point \(A\).

1. Find an equation for the tangent to \(C\) at \(A\). [3]
2. Complete the table, giving the values of \(\sqrt{x + \frac{e^x}{5}}\) to 2 decimal places.

\(x\)	0	0.5	1	1.5	2
\(\sqrt{x + \frac{e^x}{5}}\)	0.45	0.91

[2]

\includegraphics{figure_3} Figure 3 shows a sketch of the curve \(C\) with equation \(y = 3x - 2\sqrt{x}\), \(x \geqslant 0\) and the line \(l\) with equation \(y = 8x - 16\) The line cuts the curve at point \(A\) as shown in Figure 3.

1. Using algebra, find the \(x\) coordinate of point \(A\). [5]
2. \includegraphics{figure_4} The region \(R\) is shown unshaded in Figure 4. Identify the inequalities that define \(R\). [3]

A curve has equation \(y = \frac{1}{4}x^4 - x^3 - 2x^2\).

1. Find \(\frac{dy}{dx}\). [1]
2. Hence sketch the gradient function for the curve. [4]
3. Find the equation of the tangent to the curve \(y = \frac{1}{4}x^4 - x^3 - 2x^2\) at \(x = 4\). [2]

A curve has equation $$y = 2x^3 - 4x + 5$$ Find the equation of the tangent to the curve at the point \(P(2, 13)\). Write your answer in the form \(y = mx + c\), where \(m\) and \(c\) are integers to be found. Solutions relying on calculator technology are not acceptable. [5]

Curve C has equation $$y = x^3 - 7x^2 + 5x + 4$$ The point \(P(2, -6)\) lies on \(C\) Find the equation of the tangent to \(C\) at \(P\) Give your answer in the form \(y = mx + c\) where \(m\) and \(c\) are integers to be found. [3]