Find tangent line equation

2 A curve has equation \(y = 7 + 4 \ln ( 2 x + 5 )\).
Find the equation of the tangent to the curve at the point ( \(- 2,7\) ), giving your answer in the form \(y = m x + c\).

3 The equation of a curve is $$y = 6 \sin x - 2 \cos 2 x$$ Find the equation of the tangent to the curve at the point \(\left( \frac { 1 } { 6 } \pi , 2 \right)\). Give the answer in the form \(y = m x + c\), where the values of \(m\) and \(c\) are correct to 3 significant figures.

3. The curve \(C\) has the equation \(y = 2 \mathrm { e } ^ { x } - 6 \ln x\) and passes through the point \(P\) with \(x\)-coordinate 1.

1. Find an equation for the tangent to \(C\) at \(P\). The tangent to \(C\) at \(P\) meets the coordinate axes at the points \(Q\) and \(R\).
2. Show that the area of triangle \(O Q R\), where \(O\) is the origin, is \(\frac { 9 } { 3 - \mathrm { e } }\).

3 Find, in the form \(y = m x + c\), the equation of the tangent to the curve $$y = x ^ { 2 } \ln x$$ at the point with \(x\)-coordinate e.

2. The curve \(C\) has equation \(y = x ^ { \sin x } , \quad x > 0\).

1. Find the equation of the tangent to \(C\) at the point where \(x = \frac { \pi } { 2 }\).
2. Prove that this tangent touches \(C\) at infinitely many points.

6. $$\mathrm { f } ( x ) = \mathrm { e } ^ { 3 x + 1 } - 2 , \quad x \in \mathbb { R } .$$

1. State the range of f . The curve \(y = \mathrm { f } ( x )\) meets the \(y\)-axis at the point \(P\) and the \(x\)-axis at the point \(Q\).
2. Find the exact coordinates of \(P\) and \(Q\).
3. Show that the tangent to the curve at \(P\) has the equation $$y = 3 \mathrm { e } x + \mathrm { e } - 2 .$$
4. Find to 3 significant figures the \(x\)-coordinate of the point where the tangent to the curve at \(P\) meets the tangent to the curve at \(Q\).

\includegraphics{figure_1} Figure 1 shows a sketch of the curve with equation \(y = f(x)\), where $$f(x) = 10 + \ln(3x) - \frac{1}{2}e^x, \quad 0.1 \leq x \leq 3.3.$$ Given that \(f(k) = 0\),

1. show, by calculation, that \(3.1 < k < 3.2\). [2]
2. Find \(f'(x)\). [3]

The tangent to the graph at \(x = 1\) intersects the \(y\)-axis at the point \(P\).

1. 1. Find an equation of this tangent.
   2. Find the exact \(y\)-coordinate of \(P\), giving your answer in the form \(a + \ln b\). [5]

Given that \(y = \log_a x, x > 0\), where \(a\) is a positive constant,

1. 1. express \(x\) in terms of \(a\) and \(y\), [1]
   2. deduce that \(\ln x = y \ln a\). [1]
2. Show that \(\frac{dy}{dx} = \frac{1}{x \ln a}\). [2]

The curve \(C\) has equation \(y = \log_{10} x, x > 0\). The point \(A\) on \(C\) has \(x\)-coordinate \(10\). Using the result in part (b),

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

The tangent to \(C\) at \(A\) crosses the \(x\)-axis at the point \(B\).

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

\includegraphics{figure_1} Figure 1 shows a sketch of the curve with equation \(y = \text{f}(x)\), where $$\text{f}(x) = 10 + \ln(3x) - \frac{1}{2}e^x, \quad 0.1 \leq x \leq 3.3.$$ Given that f(k) = 0,

1. show, by calculation, that \(3.1 < k < 3.2\). [2]
2. Find f'(x). [3]

The tangent to the graph at \(x = 1\) intersects the \(y\)-axis at the point \(P\).

1. 1. Find an equation of this tangent.
   2. Find the exact \(y\)-coordinate of \(P\), giving your answer in the form \(a + \ln b\). [5]

Given that \(y = \log_a x\), \(x > 0\), where \(a\) is a positive constant,

1. 1. express \(x\) in terms of \(a\) and \(y\), [1]
   2. deduce that \(\ln x = y \ln a\). [1]
2. Show that \(\frac{\mathrm{d}y}{\mathrm{d}x} = \frac{1}{x \ln a}\). [2]

The curve \(C\) has equation \(y = \log_{10} x\), \(x > 0\). The point \(A\) on \(C\) has \(x\)-coordinate 10. Using the result in part (b),

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

The tangent to \(C\) at \(A\) crosses the \(x\)-axis at the point \(B\).

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