Skip to content

Home
Glossary
Chapter 1: Proof and mathematical argument
- Overview
- 1.1 Statements, assumptions and the structure of a proof
- 1.2 Proof by deduction
- 1.3 Proof by exhaustion
- 1.4 Disproof by counter example
- 1.5 Proof by contradiction
- Review
Chapter 2: Algebra and functions
- Overview
- 2.1 Laws of indices for rational exponents
- 2.2 Surds and rationalising the denominator
- 2.3 Quadratic functions, the discriminant and completing the square
- 2.4 Solving quadratic equations, including in a function of the unknown
- 2.5 Simultaneous equations by elimination and substitution
- 2.6 Linear and quadratic inequalities, and set notation
- 2.7 Polynomials, the factor theorem and algebraic division
- 2.8 Rational expressions and partial fractions
- 2.9 Graphs, the modulus function, asymptotes and intersections
- 2.10 Composite and inverse functions
- 2.11 Transformations of graphs
- 2.12 Functions in modelling: limitations and refinements
- Review
Chapter 3: Coordinate geometry
- Overview
- 3.1 Straight lines, gradients and straight-line models
- 3.2 Parallel and perpendicular lines
- 3.3 The equation of a circle and its properties
- 3.4 Parametric equations and Cartesian conversion
- 3.5 Parametric equations in modelling
- Review
Chapter 4: Sequences and series
- Overview
- 4.1 Binomial expansion for positive integer n; n! and nCr
- 4.2 Binomial expansion for rational n and approximation
- 4.3 Sequences from formulae and recurrences
- 4.4 Sigma notation for sums
- 4.5 Arithmetic sequences and series
- 4.6 Geometric sequences and series, and the sum to infinity
- 4.7 Sequences and series in modelling
- Review
Chapter 5: Trigonometry
- Overview
- 5.1 The sine and cosine rules and the area of a triangle
- 5.2 Radian measure, arc length and sector area
- 5.3 The sine, cosine and tangent functions: graphs and periodicity
- 5.4 Exact trigonometric values
- 5.5 Small-angle approximations
- 5.6 Secant, cosecant, cotangent and the inverse trigonometric functions
- 5.7 Trigonometric identities
- 5.8 Addition and double-angle formulae
- 5.9 The forms r cos(theta +/- alpha) and r sin(theta +/- alpha)
- 5.10 Solving trigonometric equations in an interval
- 5.11 Proofs with trigonometric functions and identities
- 5.12 Trigonometry in context: vectors, kinematics and forces
- Review
Chapter 6: Exponentials and logarithms
- Overview
- 6.1 The functions a^x and e^x
- 6.2 The gradient of e^kx and the exponential model
- 6.3 Logarithms and the laws of logarithms
- 6.4 Solving equations of the form a^x = b
- 6.5 Logarithmic graphs and estimating parameters
- 6.6 Exponential growth and decay in modelling
- Review
Chapter 7: Differentiation
- Overview
- 7.1 The derivative as a gradient and a rate of change
- 7.2 Differentiation from first principles
- 7.3 Differentiating x^n, e^kx, a^kx, sin kx, cos kx, tan kx and ln x
- 7.4 The second derivative, convexity and points of inflection
- 7.5 Tangents, normals, maxima, minima and stationary points
- 7.6 Increasing and decreasing functions
- 7.7 The product, quotient and chain rules
- 7.8 Connected rates of change and inverse functions
- 7.9 Implicit and parametric differentiation
- 7.10 Constructing differential equations
- Review
Chapter 8: Integration
- Overview
- 8.1 The Fundamental Theorem of Calculus and standard integrals
- 8.2 Definite integrals and the area under a curve
- 8.3 The area between two curves
- 8.4 Integration as the limit of a sum
- 8.5 Integration by substitution and by parts
- 8.6 Integrating with partial fractions
- 8.7 Separable first-order differential equations
- 8.8 Interpreting differential-equation solutions in context
- Review
Chapter 9: Numerical methods
- Overview
- 9.1 Locating roots by change of sign
- 9.2 Simple iterative methods, cobweb and staircase diagrams
- 9.3 Newton–Raphson and other recurrence relations
- 9.4 The trapezium rule and limits on the estimate
- 9.5 Numerical methods in context
- Review
Chapter 10: Vectors
- Overview
- 10.1 Vectors in two and three dimensions
- 10.2 Magnitude and direction; component and magnitude/direction form
- 10.3 Vector addition and scalar multiplication
- 10.4 Position vectors and the distance between two points
- 10.5 Vectors in pure mathematics and in context
- Review
Chapter 11: Statistical sampling and data presentation
- Overview
- 11.1 Population, sample and informal inference
- 11.2 Sampling techniques and their critique
- 11.3 Diagrams for single-variable data and the histogram
- 11.4 Scatter diagrams, regression lines and correlation
- 11.5 Measures of central tendency and variation, including standard deviation
- 11.6 Outliers, data cleaning and critiquing presentation
- 11.7 Working with the large data set
- Review
Chapter 12: Probability
- Overview
- 12.1 Mutually exclusive and independent events
- 12.2 Conditional probability and the conditional probability formula
- 12.3 Modelling with probability and critiquing assumptions
- Review
Chapter 13: Statistical distributions
- Overview
- 13.1 Discrete probability distributions and the binomial distribution
- 13.2 Calculating binomial probabilities
- 13.3 The Normal distribution as a model
- 13.4 Links to the binomial, histograms and points of inflection
- 13.5 Selecting an appropriate distribution
- Review
Chapter 14: Statistical hypothesis testing
- Overview
- 14.1 The language of hypothesis testing
- 14.2 Binomial tests for a proportion, one- and two-tailed
- 14.3 Normal tests for a mean and the significance level
- 14.4 Interpreting correlation coefficients with p-values and critical values
- Review
Chapter 15: Quantities, units and kinematics
- Overview
- 15.1 Fundamental and derived SI quantities and units
- 15.2 The language of kinematics
- 15.3 Displacement–time and velocity–time graphs
- 15.4 The constant-acceleration formulae
- 15.5 Calculus in kinematics
- 15.6 Motion under gravity and projectiles using vectors
- Review
Chapter 16: Forces, Newton's laws and moments
- Overview
- 16.1 Force and Newton's first law
- 16.2 Newton's second law and F = ma
- 16.3 Weight and motion under gravity
- 16.4 Newton's third law, equilibrium and connected particles
- 16.5 Resolving forces and coplanar equilibrium
- 16.6 Resultant forces and dynamics in a plane
- 16.7 Friction and the F <= mu R model
- 16.8 Moments in static contexts
- Review
About this book

6.1 The functions a^x and e^x — Practise

Work through these on your own. Before each question, decide whether the base lies above $1$ or between $0$ and $1$ , and note whether the question asks for an exact value or a decimal.

Both the questions and the answers below come from a set that is checked by computer every time the book is built.

Work out $2^5 \times 2^{-2}$ .
The curve $y = 7^x$ crosses the $y$ -axis at one point. Write down the $y$ -coordinate of that point.
Work out $\left(\dfrac{1}{2}\right)^{-3}$ .
Work out $16^{3/4}$ .
Use $e \approx 2.71828$ to write $e^2$ to three significant figures.
Show that $2^x = e^{x\ln 2}$ .
Work out $3^4$ and $4^3$ , and state which is larger.
Simplify $2^{x+3} \div 2^{x-1}$ .
For $y = (0.4)^x$ , work out $y$ when $x = 2$ and when $x = -1$ .
The curve $y = 3^x$ meets the line $y = 81$ once. Find the $x$ -coordinate of the meeting point.
Work out $e^0 + e^1$ to three significant figures.

Answers and concise solutions

1. The value

Answer: Add the indices: $2^5 \times 2^{-2} = 2^{5-2} = 2^3 = 8$ .

2. The $y$ -coordinate

Answer: Every curve $y = a^x$ passes through $(0, 1)$ , because $a^0 = 1$ for every positive base $a$ .

3. The value

Answer: A negative index turns the fraction over: $\left(\dfrac{1}{2}\right)^{-3} = 2^3 = 8$ .

4. The value

Answer: The power $\dfrac{3}{4}$ means the fourth root, then the cube: $16^{1/4} = 2$ , and $2^3 = 8$ .

5. The value

Answer: $e^2 = e \times e \approx 7.389056$ , which is $7.39$ to three significant figures.

6. The difference

Answer: Write the base $2$ as a power of $e$ : $2 = e^{\ln 2}$ . Then $2^x = \left(e^{\ln 2}\right)^x = e^{x\ln 2}$ , using the index law $\left(a^m\right)^n = a^{mn}$ .

7. The two values

Answer: $3^4 = 81$ and $4^3 = 64$ . So $3^4$ is larger. A larger base does not always win: the exponent matters too.

8. The value

Answer: Subtract the indices: $2^{(x+3) - (x-1)} = 2^4 = 16$ . The $x$ cancels, so the answer is a number.

9. The two values

Answer: $(0.4)^2 = 0.16 = \dfrac{4}{25}$ and $(0.4)^{-1} = \dfrac{1}{0.4} = 2.5 = \dfrac{5}{2}$ . The values fall as $x$ grows, because the base is between $0$ and $1$ .

10. The value of $x$

Answer: Write $81$ as a power of $3$ : $81 = 3^4$ . So $3^x = 3^4$ and $x = 4$ .

11. The value

Answer: $e^0 = 1$ and $e^1 = e \approx 2.71828$ . Their sum is $3.71828$ , which is $3.72$ to three significant figures.