IB Mathematics AA SL 5.1 Calculus Question Bank

[Maximum number: 2]

Let $f(x)=\frac{2 x-6}{1-x}$ , for $x \neq 1$ .

Find $\lim _{x \rightarrow \infty} f(x)$ .

valid approach
eg recognizing that $\lim _{x \rightarrow \infty}$ is related to the horizontal asymptote, table with large values of x, their y value from (a)(iii), L'Hopital's rule $\lim _{x \rightarrow \infty} f(x)=-2$

[Maximum number: 22]

The derivative of a function f is given by $f^{\prime}(x)=\frac{2 x+2}{x^{2}+2 x+2}$ , for $x \in \mathbb{R}$ .

(a)

Hence, find the values of x for which f is increasing.

[ 3 ]

$\quad x>-1($ accept $x \geq-1)$

(b)

Write down the value of x for which $f^{\prime}(x)=0$ .

[ 1 ]

$\quad x=-1$

(c)

Show that $f^{\prime \prime}(x)=\frac{-2 x^{2}-4 x}{\left(x^{2}+2 x+2\right)^{2}}$ .

[ 4 ]

attempt to use the quotient rule

f^{\prime \prime}(x)=\frac{2\left(x^{2}+2 x+2\right)-(2 x+2)(2 x+2)}{\left(x^{2}+2 x+2\right)^{2}}

Note: Award A1 for first term in numerator, A1 for second term in numerator (including the minus sign). If the numerator is correct, but the denominator is incorrect or missing, award M1A1A0.

\begin{aligned} & f^{\prime \prime}(x)=\frac{2 x^{2}+4 x+4-4 x^{2}-8 x-4}{\left(x^{2}+2 x+2\right)^{2}} \text { OR } f^{\prime \prime}(x)=\frac{2 x^{2}+4 x+4-\left(4 x^{2}+8 x+4\right)}{\left(x^{2}+2 x+2\right)^{2}} \\ & f^{\prime \prime}(x)=\frac{-2 x^{2}-4 x}{\left(x^{2}+2 x+2\right)^{2}} \end{aligned}

(d)

Hence, justify that the value of x found in part (b)(i) corresponds to a local minimum point on the graph of f.

It is given that $f(2)=3+\ln 10$ .

[ 7 ]

substituting x=-1 into $f^{\prime \prime}(x)$ , clearly leading to positive numerator A1

\begin{aligned} & f^{\prime \prime}(-1)=\left(\frac{-2+4}{1}\right)(=2) \\ & f^{\prime \prime}(-1)>0 \end{aligned}

therefore this is a local minimum point AG

Note: Do not award AOR1.

(e)

Find an expression for f(x).

[ 4 ]

recognition to integrate $f^{\prime}(x)$

\begin{aligned} & f(x)=\int \frac{2 x+2}{x^{2}+2 x+2} \mathrm{~d} x \text { OR } \int \frac{1}{u} \mathrm{~d} u \text { with } u=x^{2}+2 x+2 \\ & f(x)=\ln \left(x^{2}+2 x+2\right)(+c) \end{aligned}

using initial conditions $x=2, y=3+\ln 10$ to find c

f(2)=\ln (10)+c=3+\ln (10) \Rightarrow c=3

f(x)=\ln \left(x^{2}+2 x+2\right)+3

(f)

Find the equation of the normal to the graph of f at $(2,3+\ln 10)$ .

[ 3 ]

$f^{\prime}(2)=\frac{6}{10}$
attempt to take the negative reciprocal of their $f^{\prime}(2)$
gradient of normal $=-\frac{10}{6}$

y-(\ln 10+3)=-\frac{5}{3}(x-2) \quad\left(y=-\frac{5}{3} x+\frac{19}{3}+\ln 10\right) \quad\left(\frac{y-\ln 10-3}{x-2}=-\frac{5}{3}\right)

[Maximum number: 6]

Part of the graph of $f(x)=a x^{3}-6 x^{2}$ is shown below.

The point P lies on the graph of f. At P, x=1.

(a)

Find $f^{\prime}(x)$ .

[ 2 ]

$f^{\prime}(x)=3 a x^{2}-12 x$

Note: Award A1 for each correct term.

(b)

The graph of f has a gradient of 3 at the point P . Find the value of a.

[ 4 ]

setting their derivative equal to 3 (seen anywhere)
e.g. $\quad f^{\prime}(x)=3$
attempt to substitute x=1 into $f^{\prime}(x)$
e.g. $\quad 3 a(1)^{2}-12(1)$
correct substitution into $f^{\prime}(x)$
e.g. $\quad 3 a-12,3 a=15$
a=5
5. Note: All answers must be given in terms of m. If a candidate makes an error that means there is no m in their answer, do not award the final A1 F T mark.

METHOD 1

IB Maths AA SL 5.1 Calculus Sl Content Questions

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Question (c)

Question (d)

Question (e)

Question (f)

Question 3

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Question (b)