Sixth Term Examination Paper (STEP) Pure Mathematics: Calculus Syllabus

Pure Mathematics: Calculus syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for Pure Mathematics: Calculus in Sixth Term Examination Paper (STEP), not a summary of it.

1. 1

   Differentiation

   6 topics

   • Differentiation from first principles
   • Standard derivatives and the product, quotient and chain rules
   • Implicit and parametric differentiation
   • Derivatives of exponential, logarithmic and trig functions
     • Logarithmic differentiation
   • Stationary points, concavity and points of inflection
     • Second derivative test
     • Classifying non-standard critical points
   • Rates of change and connected rates problems
2. 2

   Integration Techniques

   6 topics

   • Integration as the reverse of differentiation; standard integrals
   • Integration by substitution
     • Trigonometric and hyperbolic substitutions
     • Choosing substitutions for awkward integrands
   • Integration by parts and reduction formulae
     • Deriving and using reduction formulae
   • Integration using partial fractions
   • Definite integrals, symmetry and the area interpretation
   • Improper integrals and convergence
3. 3

   Applications of Integration

   5 topics

   • Areas between curves and under parametric curves
   • Volumes of revolution about both axes
   • Mean value of a function over an interval
   • Arc length and surface area of revolution
   • Approximate integration (trapezium and Simpson's rules)
4. 4

   Differential Equations

   6 topics

   • Formation of differential equations from physical situations
   • First-order separable equations
   • First-order linear equations and integrating factors
   • Second-order linear equations with constant coefficients
     • Complementary function and particular integral
     • Auxiliary equation cases (real, repeated, complex roots)
   • Boundary and initial conditions
   • Modelling: growth, cooling, oscillations and damping
5. 5

   Maclaurin and Taylor Series

   4 topics

   • Maclaurin series of standard functions
   • Taylor series about a general point
   • Series solutions and combining known expansions
   • Using series for approximations and limits

Pure Mathematics: Calculus flashcards for Sixth Term Examination Paper (STEP)

23 of 54 cards from the Pure Mathematics: Calculus deck — real questions with worked answers.

1. State the definition of the derivative of $f(x)$ from first principles.

   $$f'(x)=\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}$$ provided this limit exists.

2. Using first principles, find the derivative of $f(x)=x^{n}$ for positive integer $n$ (state the result).

   Expanding $(x+h)^{n}$ by the binomial theorem gives $\frac{(x+h)^{n}-x^{n}}{h}\to nx^{n-1}$, so $f'(x)=nx^{n-1}$.

3. State the product rule for differentiating $y=u(x)\,v(x)$.

   $$\frac{dy}{dx}=u'v+uv' = u\frac{dv}{dx}+v\frac{du}{dx}$$

4. State the quotient rule for $y=\dfrac{u(x)}{v(x)}$.

   $$\frac{dy}{dx}=\frac{u'v-uv'}{v^{2}}=\frac{v\frac{du}{dx}-u\frac{dv}{dx}}{v^{2}}$$

5. State the chain rule for $y=f(g(x))$.

   $$\frac{dy}{dx}=f'(g(x))\,g'(x)=\frac{dy}{du}\cdot\frac{du}{dx}\quad(u=g(x))$$

6. Give the derivatives of $\sin x$, $\cos x$ and $\tan x$.

   $\frac{d}{dx}\sin x=\cos x$, $\frac{d}{dx}\cos x=-\sin x$, $\frac{d}{dx}\tan x=\sec^{2}x$.

7. Give the derivatives of $\sec x$, $\csc x$ and $\cot x$.

   $\frac{d}{dx}\sec x=\sec x\tan x$, $\frac{d}{dx}\csc x=-\csc x\cot x$, $\frac{d}{dx}\cot x=-\csc^{2}x$.

8. Differentiate $e^{x}$, $a^{x}$ and $\ln x$.

   $\frac{d}{dx}e^{x}=e^{x}$, $\frac{d}{dx}a^{x}=a^{x}\ln a$, $\frac{d}{dx}\ln x=\frac{1}{x}$.

9. Give the derivatives of $\arcsin x$, $\arccos x$ and $\arctan x$.

   $\frac{d}{dx}\arcsin x=\frac{1}{\sqrt{1-x^{2}}}$, $\frac{d}{dx}\arccos x=-\frac{1}{\sqrt{1-x^{2}}}$, $\frac{d}{dx}\arctan x=\frac{1}{1+x^{2}}$.

10. How do you differentiate an implicit relation $F(x,y)=0$ to find $\frac{dy}{dx}$?

    Differentiate both sides with respect to $x$, treating $y$ as a function of $x$ (so terms in $y$ pick up a factor $\frac{dy}{dx}$ via the chain rule), then solve algebraically for $\frac{dy}{dx}$.

11. For a curve given parametrically by $x=x(t)$, $y=y(t)$, give formulas for $\frac{dy}{dx}$ and $\frac{d^{2}y}{dx^{2}}$.

    $\frac{dy}{dx}=\dfrac{\dot y}{\dot x}=\dfrac{dy/dt}{dx/dt}$, and $\frac{d^{2}y}{dx^{2}}=\dfrac{\frac{d}{dt}\!\left(\frac{dy}{dx}\right)}{dx/dt}$.

12. What conditions identify a stationary point, and how does the second derivative classify it?

    Stationary where $f'(x)=0$. If $f''(x)>0$ it is a local minimum; if $f''(x)<0$ a local maximum; if $f''(x)=0$ the test is inconclusive (examine higher derivatives or the sign of $f'$).

13. Define concavity in terms of the second derivative.

    A curve is concave up (convex) where $f''(x)>0$ and concave down where $f''(x)<0$.

14. What characterises a point of inflection?

    A point where the concavity changes sign, i.e. $f''(x)=0$ and $f''$ changes sign through it. (A stationary point of inflection also has $f'(x)=0$.)

15. How are connected rates of change linked using the chain rule?

    If two quantities depend on time, $\frac{dy}{dt}=\frac{dy}{dx}\cdot\frac{dx}{dt}$; relate the variables by an equation, differentiate with respect to $t$, and substitute known rates.

16. A spherical balloon has volume $V=\frac{4}{3}\pi r^{3}$. Express $\frac{dV}{dt}$ in terms of $\frac{dr}{dt}$.

    $\frac{dV}{dt}=\frac{dV}{dr}\cdot\frac{dr}{dt}=4\pi r^{2}\,\frac{dr}{dt}$.

17. State the relationship between integration and differentiation (indefinite integral).

    Integration is the reverse of differentiation: if $\frac{d}{dx}F(x)=f(x)$ then $\int f(x)\,dx=F(x)+c$, where $c$ is an arbitrary constant.

18. Give $\int x^{n}\,dx$ (for $n\neq-1$) and $\int \frac{1}{x}\,dx$.

    $\int x^{n}\,dx=\frac{x^{n+1}}{n+1}+c$ for $n\neq-1$, and $\int\frac{1}{x}\,dx=\ln|x|+c$.

19. State $\int e^{x}\,dx$, $\int \sin x\,dx$ and $\int \sec^{2}x\,dx$.

    $\int e^{x}\,dx=e^{x}+c$, $\int\sin x\,dx=-\cos x+c$, $\int\sec^{2}x\,dx=\tan x+c$.

20. Give the standard integrals $\int\frac{dx}{a^{2}+x^{2}}$ and $\int\frac{dx}{\sqrt{a^{2}-x^{2}}}$.

    $\int\frac{dx}{a^{2}+x^{2}}=\frac{1}{a}\arctan\frac{x}{a}+c$ and $\int\frac{dx}{\sqrt{a^{2}-x^{2}}}=\arcsin\frac{x}{a}+c$.

21. Describe the method of integration by substitution.

    To find $\int f(g(x))g'(x)\,dx$, let $u=g(x)$ so $du=g'(x)\,dx$, giving $\int f(u)\,du$. For definite integrals also change the limits to values of $u$.

22. State the integration by parts formula.

    $$\int u\,\frac{dv}{dx}\,dx=uv-\int v\,\frac{du}{dx}\,dx,\qquad \int_a^b u\,dv=[uv]_a^b-\int_a^b v\,du.$$

23. What is a reduction formula and how is it typically derived?

    A recurrence expressing an integral $I_{n}$ in terms of $I_{n-1}$ (or $I_{n-2}$), usually obtained by integration by parts; repeated application reduces to a base case such as $I_{0}$ or $I_{1}$.

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