OCR A-Level Further Maths A further calculus overview quiz quiz

$x = 1$

$x = 0$

$x = \infty$

any chosen point

$x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots$

$1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + \cdots$

$1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots$

$1 + \frac{x^2}{2!} + \frac{x^4}{4!} + \frac{x^6}{6!} + \cdots$

it evaluates to a finite answer over the whole interval of integration

the integrand takes a negative value somewhere on the interval of integration

it cannot be evaluated by the method of integration by parts in closed form

a limit is infinite or the integrand is unbounded at an endpoint

$\lim_{t \to \infty}\int_a^{t} f\,dx$

$f(\infty) - f(a)$, evaluating the antiderivative at both of its limits

$\int_a^{0} f\,dx$, replacing the infinite upper limit with zero

the area of a single rectangle of height $f(a)$ and infinite width

$p < 1$

$p > 1$

$p = 1$

$p > 0$

$\pi\int y\,dx$

$2\pi\int y\,dx$

$\pi\int y^2\,dx$

$2\pi\int y^2\,dx$

$\pi\int (y_{\text{outer}} - y_{\text{inner}})^2\,dx$, squaring the difference

$\pi\int y_{\text{outer}}^2\,dx$, using only the outer curve

$\pi\int y_{\text{inner}}^2\,dx$, using only the inner curve

$\pi\int (y_{\text{outer}}^2 - y_{\text{inner}}^2)\,dx$

$e^{\int P\,dx}$

$e^{\int Q\,dx}$

$e^{Px}$

$\int P\,dx$

two distinct real roots

a repeated real root

complex roots

no roots

$Ae^{pt} + Be^{qt}$, a sum of two distinct exponentials

$(A + Bt)e^{pt}$, the repeated-root form of the solution

$e^{pt}(A\cos qt + B\sin qt)$

$A\cos pt + B\sin pt$, an undamped oscillation in $p$

heavy damping, with no oscillation

unbounded exponential growth

critically damped decay to rest

simple harmonic motion

complementary function plus particular integral

the complementary function on its own, ignoring the right-hand side

the particular integral on its own, ignoring the homogeneous part

the roots of the auxiliary equation written out as a list