1.07j Differentiate exponentials: e^(kx) and a^(kx)

1. The number of bacteria on a surface is being monitored.

The number of bacteria, \(N\), on the surface, \(t\) hours after monitoring began is modelled by the equation $$\log _ { 10 } N = 0.35 t + 2$$ Use the equation of the model to answer parts (a) to (c).

1. Find the initial number of bacteria on the surface.
2. Show that the equation of the model can be written in the form $$N = a b ^ { t }$$ where \(a\) and \(b\) are constants to be found. Give the value of \(b\) to 2 decimal places.
3. Hence find the rate of growth of bacteria on the surface exactly 5 hours after monitoring began.

8. \begin{figure}[h] \end{figure} The heart rate of a horse is being monitored.
The heart rate \(H\), measured in beats per minute (bpm), is modelled by the equation $$H = 32 + 40 \mathrm { e } ^ { - 0.2 t } - 20 \mathrm { e } ^ { - 0.9 t }$$ where \(t\) minutes is the time after monitoring began.
Figure 4 is a sketch of \(H\) against \(t\). \section*{Use the equation of the model to answer parts (a) to (e).}

1. State the initial heart rate of the horse. In the long term, the heart rate of the horse approaches \(L \mathrm { bpm }\).
2. State the value of \(L\). The heart rate of the horse reaches its maximum value after \(T\) minutes.
3. Find the value of \(T\), giving your answer to 3 decimal places.
   (Solutions based entirely on calculator technology are not acceptable.) The heart rate of the horse is 37 bpm after \(M\) minutes.
4. Show that \(M\) is a solution of the equation $$t = 5 \ln \left( \frac { 8 } { 1 + 4 \mathrm { e } ^ { - 0.9 t } } \right)$$ Using the iteration formula $$t _ { n + 1 } = 5 \ln \left( \frac { 8 } { 1 + 4 \mathrm { e } ^ { - 0.9 t _ { n } } } \right) \quad \text { with } \quad t _ { 1 } = 10$$
5. 1. find, to 4 decimal places, the value of \(t _ { 2 }\)
   2. find, to 4 decimal places, the value of \(M\)

4. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of the curve \(C\) with equation $$3 x ^ { 2 } + 2 y ^ { 2 } - 4 x y + 8 ^ { x } - 11 = 0$$ The point \(P\) has coordinates ( 1,2 ).

1. Verify that \(P\) lies on \(C\).
2. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\). The normal to \(C\) at \(P\) crosses the \(x\)-axis at a point \(Q\).
3. Find the \(x\) coordinate of \(Q\), giving your answer in the form \(a + b \ln 2\) where \(a\) and \(b\) are integers.

10 A curve has equation \(y = \frac { \mathrm { e } ^ { x } } { x ^ { 2 } }\). Show that

1. the gradient of the curve at \(x = 1\) is - e ,
2. there is a stationary point at \(x = 2\) and determine its nature.

10 A curve has equation $$y = \mathrm { e } ^ { a x } \cos b x$$ where \(a\) and \(b\) are constants.

1. Show that, at any stationary points on the curve, \(\tan b x = \frac { a } { b }\).
2. \includegraphics[max width=\textwidth, alt={}, center]{1c957cfe-bead-41d9-8985-479e876e1616-4_620_896_959_333} Values of related quantities \(x\) and \(y\) were measured in an experiment and plotted on a graph of \(y\) against \(x\), as shown in the diagram. Two of the points, labelled \(A\) and \(B\), have coordinates \(( 0,1 )\) and \(( 0.2 , - 0.8 )\) respectively. A third point labelled C has coordinates ( \(0.3,0.04\) ). Attempts were then made to find the equation of a curve which fitted closely to these three points, and two models were proposed. In the first model the equation is \(y = \mathrm { e } ^ { - x } \cos 15 x\).
   In the second model the equation is \(y = f \cos ( \lambda x ) + \mathrm { g }\), where the constants \(f , \lambda\), and \(g\) are chosen to give a maximum precisely at the point \(A ( 0,1 )\) and a minimum precisely at the point \(B ( 0.2 , - 0.8 )\). By calculating suitable values evaluate the suitability of the two models.

10 A curve has equation $$y = \mathrm { e } ^ { a x } \cos b x$$ where \(a\) and \(b\) are constants.

1. Show that, at any stationary points on the curve, \(\tan b x = \frac { a } { b }\).
2. \includegraphics[max width=\textwidth, alt={}, center]{8a0a6e46-99cf-4217-93ad-5ed6e9d7c4ef-4_624_897_1062_342} Values of related quantities \(x\) and \(y\) were measured in an experiment and plotted on a graph of \(y\) against \(x\), as shown in the diagram. Two of the points, labelled \(A\) and \(B\), have coordinates \(( 0,1 )\) and \(( 0.2 , - 0.8 )\) respectively. A third point labelled \(C\) has coordinates \(( 0.3,0.04 )\). Attempts were then made to find the equation of a curve which fitted closely to these three points, and two models were proposed. In the first model the equation is \(y = \mathrm { e } ^ { - x } \cos 15 x\).
   In the second model the equation is \(y = f \cos ( \lambda x ) + g\), where the constants \(f , \lambda\), and \(g\) are chosen to give a maximum precisely at the point \(A ( 0,1 )\) and a minimum precisely at the point \(B ( 0.2 , - 0.8 )\). By calculating suitable values evaluate the suitability of the two models.

9 A curve has equation $$y = \mathrm { e } ^ { 3 x } - 5 \mathrm { e } ^ { 2 x } + 8 \mathrm { e } ^ { x }$$

1. Find the exact coordinates of the stationary points of \(y\).
2. Determine the range of values of \(x\) for which $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } > 0$$
3. Determine the nature of the stationary points on the curve.

\includegraphics{figure_3} The diagram shows the curve with equation \(y = 8e^{-x} - e^{2x}\). The curve crosses the y-axis at the point A and the x-axis at the point B. The shaded region is bounded by the curve and the two axes.

1. Find the gradient of the curve at A. [3]
2. Show that the x-coordinate of B is \(\ln 2\) and hence find the area of the shaded region. [5]

\includegraphics{figure_6} The diagram shows the curve with equation \(y = \frac{4e^{2x} + 9}{e^x + 2}\). The curve has a minimum point \(M\) and crosses the \(y\)-axis at the point \(P\).

1. Find the exact value of the gradient of the curve at \(P\). [4]
2. Find the exact coordinates of \(M\). [4]

A curve has parametric equations $$x = \frac{e^{2t} - 2}{e^{2t} + 1}, \quad y = e^{3t} + 1.$$

1. Find an expression for \(\frac{dy}{dx}\) in terms of \(t\). [4]
2. Find the exact gradient of the curve at the point where the curve crosses the \(y\)-axis. [3]

\includegraphics{figure_4} The diagram shows the curve \(y = e^x + 4e^{-2x}\) and its minimum point \(M\).

1. Show that the \(x\)-coordinate of \(M\) is \(\ln 2\). [3]
2. The region shaded in the diagram is enclosed by the curve and the lines \(x = 0\), \(x = \ln 2\) and \(y = 0\). Use integration to show that the area of the shaded region is \(\frac{5}{2}\). [4]

\includegraphics{figure_6} The diagram shows the curve \(y = (4 - x)e^x\) and its maximum point \(M\). The curve cuts the \(x\)-axis at \(A\) and the \(y\)-axis at \(B\).

1. Write down the coordinates of \(A\) and \(B\). [2]
2. Find the \(x\)-coordinate of \(M\). [4]
3. The point \(P\) on the curve has \(x\)-coordinate \(p\). The tangent to the curve at \(P\) passes through the origin \(O\). Calculate the value of \(p\). [5]

A curve has parametric equations $$x = t + \ln(t + 1), \quad y = 3te^{2t}.$$

1. Find the equation of the tangent to the curve at the origin. [5]
2. Find the coordinates of the stationary point, giving each coordinate correct to 2 decimal places. [4]

Find the exact coordinates of the stationary point of the curve \(y = e^{2x} \sin 2x\) for \(0 \leqslant x < \frac{1}{2}\pi\). [5]

Let \(\text{f}(x) = \frac{e^{2x} + 1}{e^{2x} - 1}\), for \(x > 0\).

1. The equation \(x = \text{f}(x)\) has one root, denoted by \(a\). Verify by calculation that \(a\) lies between 1 and 1.5. [2]
2. Use an iterative formula based on the equation in part (a) to determine \(a\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places. [3]
3. Find f\('(x)\). Hence find the exact value of \(x\) for which f\('(x) = -8\). [6]

A balloon in the shape of a sphere has volume \(V\) and radius \(r\). Air is pumped into the balloon at a constant rate of \(40\pi\) starting when time \(t = 0\) and \(r = 0\). At the same time, air begins to flow out of the balloon at a rate of \(0.8\pi r\). The balloon remains a sphere at all times.

1. Show that \(r\) and \(t\) satisfy the differential equation $$\frac{dr}{dt} = \frac{50 - r}{5r^2}.$$ [3]
2. Find the quotient and remainder when \(5r^2\) is divided by \(50 - r\). [3]
3. Solve the differential equation in part (a), obtaining an expression for \(t\) in terms of \(r\). [6]
4. Find the value of \(t\) when the radius of the balloon is 12. [1]

\includegraphics{figure_5} The diagram shows the curve \(y = e^{-x} - e^{-2x}\) and its maximum point \(M\). The \(x\)-coordinate of \(M\) is denoted by \(p\).

1. Find the exact value of \(p\). [4]
2. Show that the area of the shaded region bounded by the curve, the \(x\)-axis and the line \(x = p\) is equal to \(\frac{1}{4}\). [4]

The parametric equations of a curve are $$x = e^{-t}\cos t, \quad y = e^{-t}\sin t.$$ Show that \(\frac{dy}{dx} = \tan(t - \frac{1}{4}\pi)\). [6]

Given that \(a\) is a constant, prove by mathematical induction that, for every positive integer \(n\), $$\frac{\mathrm{d}^n}{\mathrm{d}x^n}(xe^{ax}) = na^{n-1}e^{ax} + a^n xe^{ax}.$$ [6]

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_2} Figure 2 shows part of the curve \(C\) with equation \(y = f(x)\), where $$f(x) = 0.5e^x - x^2.$$ 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]

\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]

The curve \(C\) with equation \(y = p + qe^x\), where \(p\) and \(q\) are constants, passes through the point \((0, 2)\). At the point \(P(\ln 2, p + 2q)\) on \(C\), the gradient is \(5\).

1. Find the value of \(p\) and the value of \(q\). [5]

The normal to \(C\) at \(P\) crosses the \(x\)-axis at \(L\) and the \(y\)-axis at \(M\).

1. Show that the area of \(\triangle OLM\), where \(O\) is the origin, is approximately \(53.8\). [5]

\includegraphics{figure_3} Figure 3 shows a sketch of part of the curve \(C\) with equation $$y = 3^x$$ The point \(P\) lies on \(C\) and has coordinates \((2, 9)\). The line \(l\) is a tangent to \(C\) at \(P\). The line \(l\) cuts the \(x\)-axis at the point \(Q\).

1. Find the exact value of the \(x\) coordinate of \(Q\). [4]

The finite region \(R\), shown shaded in Figure 3, is bounded by the curve \(C\), the \(x\)-axis, the \(y\)-axis and the line \(l\). This region \(R\) is rotated through \(360°\) about the \(x\)-axis.

1. Use integration to find the exact value of the volume of the solid generated. Give your answer in the form \(\frac{p}{q}\) where \(p\) and \(q\) are exact constants. [You may assume the formula \(V = \frac{1}{3}\pi r^2 h\) for the volume of a cone.] [6]