1.06a Exponential function: a^x and e^x graphs and properties

3 A curve has equation \(\mathrm { e } ^ { 2 x } \cos 2 y + \sin y = 1\).
Find the exact gradient of the curve at the point \(\left( 0 , \frac { 1 } { 6 } \pi \right)\).

5 The polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = 2 x ^ { 3 } + a x ^ { 2 } - 3 x - 4$$ where \(a\) is a constant. It is given that ( \(x - 4\) ) is a factor of \(\mathrm { p } ( x )\).

1. Find the value of \(a\) and hence factorise \(\mathrm { p } ( x )\).
2. Show that the equation \(\mathrm { p } \left( \mathrm { e } ^ { 3 y } \right) = 0\) has only one real root and find its exact value.

6 \includegraphics[max width=\textwidth, alt={}, center]{1b9c6b41-69dd-4132-92c7-9507cbd741dd-08_542_661_269_731} The diagram shows the curve \(y = 3 \mathrm { e } ^ { 2 x - 1 }\). The shaded region is bounded by the curve and the lines \(x = a , x = a + 1\) and \(y = 0\), where \(a\) is a constant. It is given that the area of the shaded region is 120 square units.

1. Show that \(a = \frac { 1 } { 2 } \ln \left( 80 + \mathrm { e } ^ { 2 a - 1 } \right) - \frac { 1 } { 2 }\).
2. Use an iterative formula, based on the equation in part (a), to find the value of \(a\) correct to 3 significant figures. Give the result of each iteration to 5 significant figures.

6 \includegraphics[max width=\textwidth, alt={}, center]{712be8e6-e1e9-4662-b1f1-51c39c2c9df1-08_542_661_269_731} The diagram shows the curve \(y = 3 \mathrm { e } ^ { 2 x - 1 }\). The shaded region is bounded by the curve and the lines \(x = a , x = a + 1\) and \(y = 0\), where \(a\) is a constant. It is given that the area of the shaded region is 120 square units.

1. Show that \(a = \frac { 1 } { 2 } \ln \left( 80 + \mathrm { e } ^ { 2 a - 1 } \right) - \frac { 1 } { 2 }\).
2. Use an iterative formula, based on the equation in part (a), to find the value of \(a\) correct to 3 significant figures. Give the result of each iteration to 5 significant figures.

3 It is given that \(\int _ { 0 } ^ { a } \left( 3 \mathrm { e } ^ { 2 x } - 1 \right) \mathrm { d } x = 12\), where \(a\) is a positive constant.

1. Show that \(a = \frac { 1 } { 2 } \ln \left( 9 + \frac { 2 } { 3 } a \right)\).
2. Use an iterative formula, based on the equation in (a), to find the value of \(a\) correct to 4 significant figures. Use an initial value of 1 and give the result of each iteration to 6 significant figures. [3]

5 \includegraphics[max width=\textwidth, alt={}, center]{3966e088-0a2f-434a-94fc-40765cd157a7-06_376_848_269_644} The diagram shows the curve with parametric equations $$x = 4 \mathrm { e } ^ { 2 t } , \quad y = 5 \mathrm { e } ^ { - t } \cos 2 t$$ for \(- \frac { 1 } { 4 } \pi \leqslant t \leqslant \frac { 1 } { 4 } \pi\). The curve has a maximum point \(M\).

1. Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\).
2. Find the coordinates of \(M\), giving each coordinate correct to 3 significant figures.

4

1. \includegraphics[max width=\textwidth, alt={}, center]{4ce3208e-8ceb-4848-a9c7-fcda166319f4-05_753_944_278_630} The diagram shows the graph of \(y = 3 - \mathrm { e } ^ { - \frac { 1 } { 2 } x }\).
   On the diagram, sketch the graph of \(y = | 5 x - 4 |\), and show that the equation \(3 - e ^ { - \frac { 1 } { 2 } x } = | 5 x - 4 |\) has exactly two real roots. It is given that the two roots of \(3 - \mathrm { e } ^ { - \frac { 1 } { 2 } x } = | 5 x - 4 |\) are denoted by \(\alpha\) and \(\beta\), where \(\alpha < \beta\).
2. Show by calculation that \(\alpha\) lies between 0.36 and 0.37 .
3. Use the iterative formula \(x _ { n + 1 } = \frac { 1 } { 5 } \left( 7 - \mathrm { e } ^ { - \frac { 1 } { 2 } x _ { n } } \right)\) to find \(\beta\) correct to 4 significant figures. Give the result of each iteration to 6 significant figures.

5 \includegraphics[max width=\textwidth, alt={}, center]{4ce3208e-8ceb-4848-a9c7-fcda166319f4-06_526_947_276_591} The diagram shows the curve with equation \(y = \mathrm { e } ^ { - \frac { 1 } { 2 } x } \left( x ^ { 2 } - 5 x + 4 \right)\). The curve crosses the \(x\)-axis at the points \(A\) and \(B\), and has a maximum at the point \(C\).

1. Find the exact gradient of the curve at \(B\).
2. Find the exact coordinates of \(C\).

4

1. \includegraphics[max width=\textwidth, alt={}, center]{a1ea242a-c7f4-46b0-b4b8-bd13b3880557-05_753_944_278_630} The diagram shows the graph of \(y = 3 - \mathrm { e } ^ { - \frac { 1 } { 2 } x }\).
   On the diagram, sketch the graph of \(y = | 5 x - 4 |\), and show that the equation \(3 - e ^ { - \frac { 1 } { 2 } x } = | 5 x - 4 |\) has exactly two real roots. It is given that the two roots of \(3 - \mathrm { e } ^ { - \frac { 1 } { 2 } x } = | 5 x - 4 |\) are denoted by \(\alpha\) and \(\beta\), where \(\alpha < \beta\).
2. Show by calculation that \(\alpha\) lies between 0.36 and 0.37 .
3. Use the iterative formula \(x _ { n + 1 } = \frac { 1 } { 5 } \left( 7 - \mathrm { e } ^ { - \frac { 1 } { 2 } x _ { n } } \right)\) to find \(\beta\) correct to 4 significant figures. Give the result of each iteration to 6 significant figures.

3

1. Sketch on the same diagram the graphs of \(y = | 3 x - 8 |\) and \(y = 5 - x\).
2. Solve the inequality \(| 3 x - 8 | < 5 - x\).
3. Hence determine the largest integer \(N\) satisfying the inequality \(\left| 3 e ^ { 0.1 N } - 8 \right| < 5 - e ^ { 0.1 N }\).

3 \includegraphics[max width=\textwidth, alt={}, center]{8beee722-7f86-454a-bc36-27e83f1483fd-04_684_455_260_845} The diagram shows the curve \(y = 2 + \mathrm { e } ^ { - 2 x }\). The curve crosses the \(y\)-axis at the point \(A\), and the point \(B\) on the curve has \(x\)-coordinate 1 . The shaded region is bounded by the curve and the line segment \(A B\). Find the exact area of the shaded region.

3 \includegraphics[max width=\textwidth, alt={}, center]{b4a4082c-f3cd-47c5-8673-680dae9a22bd-04_684_455_260_845} The diagram shows the curve \(y = 2 + \mathrm { e } ^ { - 2 x }\). The curve crosses the \(y\)-axis at the point \(A\), and the point \(B\) on the curve has \(x\)-coordinate 1 . The shaded region is bounded by the curve and the line segment \(A B\). Find the exact area of the shaded region.

6

1. By sketching a suitable pair of graphs on the same diagram, show that the equation $$\ln x = 2 \mathrm { e } ^ { - x }$$ has exactly one root.
2. Verify by calculation that the root lies between 1.5 and 1.6.
3. Show that if a sequence of values given by the iterative formula $$x _ { n + 1 } = \mathrm { e } ^ { 2 \mathrm { e } ^ { - x _ { n } } }$$ converges, then it converges to the root of the equation in part (a).
4. Use the iterative formula in part (c) to determine the root correct to 3 significant figures. Give the result of each iteration to 5 significant figures.

3 \includegraphics[max width=\textwidth, alt={}, center]{b104e2a7-06c8-4e2e-a4f9-5095ad56897a-04_652_392_274_872} The diagram shows the curve with equation \(y = 6 \mathrm { e } ^ { - \frac { 1 } { 2 } x }\). The points on the curve with \(x\)-coordinates 0 and 2 are denoted by \(A\) and \(B\) respectively. The shaded region is enclosed by the curve, the line through \(A\) parallel to the \(x\)-axis and the line through \(B\) parallel to the \(y\)-axis.

1. Find the exact gradient of the curve at \(B\).
2. Find the exact area of the shaded region.

7 \includegraphics[max width=\textwidth, alt={}, center]{9d93ad8c-0a22-4de7-8342-387606e4e510-3_584_675_945_735} The diagram shows the part of the curve \(y = \mathrm { e } ^ { x } \cos x\) for \(0 \leqslant x \leqslant \frac { 1 } { 2 } \pi\). The curve meets the \(y\)-axis at the point \(A\). The point \(M\) is a maximum point.

1. Write down the coordinates of \(A\).
2. Find the \(x\)-coordinate of \(M\).
3. Use the trapezium rule with three intervals to estimate the value of $$\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } e ^ { x } \cos x d x$$ giving your answer correct to 2 decimal places.
4. State, with a reason, whether the trapezium rule gives an under-estimate or an over-estimate of the true value of the integral in part (iii).

7

1. By sketching a suitable pair of graphs, show that the equation $$\mathrm { e } ^ { 2 x } = 2 - x$$ has only one root.
2. Verify by calculation that this root lies between \(x = 0\) and \(x = 0.5\).
3. Show that, if a sequence of values given by the iterative formula $$x _ { n + 1 } = \frac { 1 } { 2 } \ln \left( 2 - x _ { n } \right)$$ converges, then it converges to the root of the equation in part (i).
4. Use this iterative formula, with initial value \(x _ { 1 } = 0.25\), to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
5. By differentiating \(\frac { \cos x } { \sin x }\), show that if \(y = \cot x\) then \(\frac { \mathrm { d } y } { \mathrm {~d} x } = - \operatorname { cosec } ^ { 2 } x\).
6. By expressing \(\cot ^ { 2 } x\) in terms of \(\operatorname { cosec } ^ { 2 } x\) and using the result of part (i), show that $$\int _ { \frac { 1 } { 4 } \pi } ^ { \frac { 1 } { 2 } \pi } \cot ^ { 2 } x \mathrm {~d} x = 1 - \frac { 1 } { 4 } \pi$$
7. Express \(\cos 2 x\) in terms of \(\sin ^ { 2 } x\) and hence show that \(\frac { 1 } { 1 - \cos 2 x }\) can be expressed as \(\frac { 1 } { 2 } \operatorname { cosec } ^ { 2 } x\). Hence, using the result of part (i), find $$\int \frac { 1 } { 1 - \cos 2 x } \mathrm {~d} x$$

7

1. By sketching a suitable pair of graphs, show that the equation $$\mathrm { e } ^ { 2 x } = 14 - x ^ { 2 }$$ has exactly two real roots.
2. Show by calculation that the positive root lies between 1.2 and 1.3.
3. Show that this root also satisfies the equation $$x = \frac { 1 } { 2 } \ln \left( 14 - x ^ { 2 } \right) .$$
4. Use an iteration process based on the equation in part (iii), with a suitable starting value, to find the root correct to 2 decimal places. Give the result of each step of the process to 4 decimal places.
5. Express \(4 \sin \theta - 6 \cos \theta\) in the form \(R \sin ( \theta - \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\). Give the exact value of \(R\) and the value of \(\alpha\) correct to 2 decimal places.
6. Solve the equation \(4 \sin \theta - 6 \cos \theta = 3\) for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).
7. Find the greatest and least possible values of \(( 4 \sin \theta - 6 \cos \theta ) ^ { 2 } + 8\) as \(\theta\) varies.

6

1. By sketching a suitable pair of graphs, show that the equation $$3 \mathrm { e } ^ { x } = 8 - 2 x$$ has only one root.
2. Verify by calculation that this root lies between \(x = 0.7\) and \(x = 0.8\).
3. Show that this root also satisfies the equation $$x = \ln \left( \frac { 8 - 2 x } { 3 } \right)$$
4. Use the iterative formula \(x _ { n + 1 } = \ln \left( \frac { 8 - 2 x _ { n } } { 3 } \right)\) to determine this root correct to 3 decimal places. Give the result of each iteration to 5 decimal places.

1

1. Solve the equation \(| x + 2 | = | x - 13 |\).
2. Hence solve the equation \(\left| 3 ^ { y } + 2 \right| = \left| 3 ^ { y } - 13 \right|\), giving your answer correct to 3 significant figures.

1 Solve the equation \(3 \mathrm { e } ^ { 2 x } - 82 \mathrm { e } ^ { x } + 27 = 0\), giving your answers in the form \(k \ln 3\).

3 Without using a calculator, find the exact value of \(\int _ { 0 } ^ { 2 } 4 \mathrm { e } ^ { - x } \left( \mathrm { e } ^ { 3 x } + 1 \right) \mathrm { d } x\).

6 It is given that \(\int _ { 0 } ^ { a } \left( 1 + \mathrm { e } ^ { \frac { 1 } { 2 } x } \right) ^ { 2 } \mathrm {~d} x = 10\), where \(a\) is a positive constant.

1. Show that \(a = 2 \ln \left( \frac { 15 - a } { 4 + \mathrm { e } ^ { \frac { 1 } { 2 } a } } \right)\).
2. Use the equation in part (i) to show by calculation that \(1.5 < a < 1.6\).
3. Use an iterative formula based on the equation in part (i) to find the value of \(a\) correct to 3 significant figures. Give the result of each iteration to 5 significant figures.

5 In a certain chemical process a substance \(A\) reacts with another substance \(B\). The masses in grams of \(A\) and \(B\) present at time \(t\) seconds after the start of the process are \(x\) and \(y\) respectively. It is given that \(\frac { \mathrm { d } y } { \mathrm {~d} t } = - 0.6 x y\) and \(x = 5 \mathrm { e } ^ { - 3 t }\). When \(t = 0 , y = 70\).

1. Form a differential equation in \(y\) and \(t\). Solve this differential equation and obtain an expression for \(y\) in terms of \(t\).
2. The percentage of the initial mass of \(B\) remaining at time \(t\) is denoted by \(p\). Find the exact value approached by \(p\) as \(t\) becomes large.

6

1. By sketching a suitable pair of graphs, show that the equation $$5 \mathrm { e } ^ { - x } = \sqrt { } x$$ has one root.
2. Show that, if a sequence of values given by the iterative formula $$x _ { n + 1 } = \frac { 1 } { 2 } \ln \left( \frac { 25 } { x _ { n } } \right)$$ converges, then it converges to the root of the equation in part (i).
3. Use this iterative formula, with initial value \(x _ { 1 } = 1\), to calculate the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

6 The variables \(x\) and \(y\) satisfy the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = k y ^ { 3 } \mathrm { e } ^ { - x }$$ where \(k\) is a constant. It is given that \(y = 1\) when \(x = 0\), and that \(y = \sqrt { } \mathrm { e }\) when \(x = 1\). Solve the differential equation, obtaining an expression for \(y\) in terms of \(x\).