Let $f(x) = 3x^2 + x$. Use the definition $$f'(a) = rac{f(a + h) - f(a)}{h}$$ to find the derivative of $f(x)$ at the point $x = a$. (i) Find $\int \frac{e^x}{1+e^x} dx$. (ii) Find $\int_0^2 3 \cos x dx$. The letters A, E, I, O, and U are vowels. (i) How many arrangements of the letters in the word ALGE BRAIC are possible? (ii) How many arrangements of the letters in the word ALGE BRAIC are possible if the vowels must occupy the 2nd, 3rd, 5th and 8th positions? (d) Find the term independent of $x$ in the binomial expansion of $(x^2 - 1)^9$.

SSCE HSC Mathematics Extension 1 Integration of sin^2x and cos^2x: Let $f(x) = 3x^2 + x$. Use the d

Let $f(x) = 3x^2 + x$ - HSC - SSCE Mathematics Extension 1 - Question 2 - 2001 - Paper 1

Question 2

Let $f(x) = 3x^2 + x$. Use the definition $$f'(a) = rac{f(a + h) - f(a)}{h}$$ to find the derivative of $f(x)$ at the point $x = a$. (i) Find $\int \frac{e^x}{1+... show full transcript

Worked Solution & Example Answer:Let $f(x) = 3x^2 + x$ - HSC - SSCE Mathematics Extension 1 - Question 2 - 2001 - Paper 1

Step 1

Let $f(x) = 3x^2 + x$. Use the definition to find the derivative of $f(x)$ at the point $x = a$.

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Answer

To find the derivative using the limit definition:

$f'(a) = \lim_{h \to 0} \frac{f(a + h) - f(a)}{h}$

Calculate:

$f(a + h) = 3(a + h)^2 + (a + h) = 3(a^2 + 2ah + h^2) + a + h = 3a^2 + 6ah + 3h^2 + a + h$ .
$f(a) = 3a^2 + a$ .
Therefore,

$f'(a) = \lim_{h \to 0} \frac{(3a^2 + 6ah + 3h^2 + a + h) - (3a^2 + a)}{h} = \lim_{h \to 0} \frac{6ah + 3h^2 + h}{h}$

This simplifies to:

$f'(a) = \lim_{h \to 0} (6a + 3h + 1) = 6a + 1.$

Step 2

Find $\int \frac{e^x}{1+e^x} dx$.

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Answer

To evaluate this integral, use the substitution $u = 1 + e^x$ , hence $du = e^x dx$ .

Then the integral becomes:

$\int \frac{1}{u} du = \ln |u| + C = \ln |1 + e^x| + C.$

Step 3

Find $\int_0^2 3 \cos x dx$.

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Answer

Evaluate this integral as follows:

$\int 3 \cos x dx = 3 \sin x + C.$

Apply the limits:

$\left[3 \sin x\right]_0^2 = 3 \sin(2) - 3 \sin(0) = 3 \sin(2).$

Step 4

How many arrangements of the letters in the word ALGE BRAIC are possible?

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Answer

The word 'ALGE BRAIC' has 9 letters with A occurring twice and I occurring twice.

The formula for arrangements is:

$\frac{n!}{n_1! \cdot n_2!},$ where $n$ is the total number of letters, $n_1$ and $n_2$ are the counts of respective repeated letters.

Thus:

$\frac{9!}{2! \cdot 2!} = \frac{362880}{4} = 90720.$

Step 5

How many arrangements of the letters in the word ALGE BRAIC are possible if the vowels must occupy the 2nd, 3rd, 5th and 8th positions?

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Answer

The vowels A, A, E, I need to occupy positions 2, 3, 5, and 8. The arrangements of the vowels:

$\frac{4!}{2!} = 12.$

The remaining consonants L, G, B, R, C can occupy the remaining 5 positions:

$5! = 120.$

Therefore, the total arrangements is:

$12 \times 120 = 1440.$

Step 6

Find the term independent of $x$ in the binomial expansion of $(x^2 - 1)^9$.

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Answer

Using the binomial theorem, the general term is given by:

$T_k = \binom{n}{k} (x^2)^{n-k} (-1)^k = \binom{9}{k} x^{2(n-k)}(-1)^k.$

We seek the term where the exponent of $x$ is zero:

$2(n-k) = 0 \Rightarrow n-k = 0 \Rightarrow k = 9.$

Thus, we find the term:

$inom{9}{9}(-1)^9 = -1.$

Let $f(x) = 3x^2 + x$ - HSC - SSCE Mathematics Extension 1 - Question 2 - 2001 - Paper 1

Worked Solution & Example Answer:Let $f(x) = 3x^2 + x$ - HSC - SSCE Mathematics Extension 1 - Question 2 - 2001 - Paper 1

Let $f(x) = 3x^2 + x$. Use the definition to find the derivative of $f(x)$ at the point $x = a$.

Find $\int \frac{e^x}{1+e^x} dx$.

Find $\int_0^2 3 \cos x dx$.

How many arrangements of the letters in the word ALGE BRAIC are possible?

How many arrangements of the letters in the word ALGE BRAIC are possible if the vowels must occupy the 2nd, 3rd, 5th and 8th positions?

Find the term independent of $x$ in the binomial expansion of $(x^2 - 1)^9$.

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