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Exponential functions and e questions

Put x in the exponent and everything changes. Equal steps in x now multiply y instead of adding to it. Among all the possible bases one is special, because the curve of e to the x climbs at a rate equal to its own height, and that single property is why the same number keeps turning up in banking, biology and radioactive decay.

7 original questions · 21 marks · the exponential functions and e notes · Exponentials and logarithms

Every question here is written for this library rather than taken from a past paper. Write your answer out before opening the worked one: the answers award marks point by point, and the marks are easier to see when you have something of your own to compare against.

• 3 marks · warm-up

  For the curve y = 3^x, write down the y-intercept, the value of y at x = 2, and the equation of the asymptote.

  Worked answer

  The intercept is (0, 1), since 3⁰ = 1. At x = 2, y = 9. The asymptote is y = 0, the x-axis, which the curve approaches on the left without ever touching. B1 B1 B1 for the three answers. Every curve y = a^x passes through (0, 1) whatever a is, so that intercept is worth remembering rather than working out.

• 2 marks · explain · warm-up

  Explain why the equation 5^x = 0 has no solution.

  Worked answer

  Every exponential is positive everywhere: 5^x takes every value above zero and no others. The curve hugs the x-axis as x heads left but never reaches it, so zero is approached, not achieved. B1 for the range being strictly positive, B1 for the asymptote argument.

• 3 marks

  For the curve y = e^4x, write down the gradient function, the gradient at x = 0, and the gradient at x = 0.5 to 3 significant figures.

  Worked answer

  Differentiating e^kx multiplies by k, so the gradient function is 4e^4x. At x = 0 the gradient is 4e⁰ = 4, and at x = 0.5 it is 4e² = 29.6 to 3 significant figures, four times the height there. B1 for the gradient function, B1 for 4, B1 for 29.6. The factor of 4 is the only mark at risk; writing e^4x unchanged is the error the chain rule exists to prevent.

• 3 marks

  For the curve y = e^−2x, write down the gradient function and the value of y at x = 1, and describe the curve's behaviour as x grows.

  Worked answer

  The gradient function is −2e^−2x, negative everywhere: the curve falls at every point. At x = 1, y = e⁻² = 0.135 to 3 significant figures. As x grows the curve decays towards the asymptote y = 0, steeply at first and ever more gently. B1 for the gradient function, B1 for 0.135, B1 for the decay to the asymptote.

• 3 marks

  A population of bacteria triples every hour, starting from 500. Write down a model for the population P after t hours, and find the population after 4 hours.

  Worked answer

  P = 500 × 3^t, since each unit step in t multiplies the population by the same ratio 3. After 4 hours, P = 500 × 81 = 40 500. B1 for the model, M1 for substituting t = 4, A1 for the value after four hours. The starting value belongs in front and the growth factor in the base, so P = 500t³ or P = 3 × 500^t both fail the first check, which is that P = 500 at t = 0.

• 4 marks · stretch

  For y = 5e^0.2x, show that the gradient at every point equals 0.2y, and find the height and gradient at x = 5, each to 3 significant figures.

  Worked answer

  The gradient function is 5 × 0.2e^0.2x = 0.2 × 5e^0.2x = 0.2y: the rate of change is proportional to the amount, with constant 0.2. At x = 5 the height is 5e = 13.6 and the gradient is 0.2 × 5e = e = 2.72. M1 for differentiating, A1 for the link to 0.2y, A1 for 13.6, A1 for 2.72. This proportionality is the property that makes e the modelling base.

• 3 marks · stretch

  The curves y = 2^x and y = 3^x have gradients of about 0.69 and 1.10 at their common point (0, 1). Explain what this suggests about the number e, and state the defining property of y = e^x.

  Worked answer

  A gradient of exactly 1 at (0, 1) must belong to some base between 2 and 3, and that base is e = 2.718…. The defining property is that y = e^x is its own gradient function: at every point the gradient equals the height, and (0, 1) is only the most convenient place to notice it. B1 for the base lying between 2 and 3, B1 for e = 2.718, B1 for the self-derivative property.

The same practice on paper: the printable workbook for this topic, questions and a worked answer book.

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