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 $$f'(a) = ext{lim}_{h \to 0} \frac{f(a + h) - f(a)}{h}$$ to find the derivative of $f(x)$ at the point $x = a$. (i) Fin... 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 $x = a$.

96%

114 rated

Answer

To find the derivative using the limit definition, we substitute into the formula:

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

Calculating $f(a + h)$ :

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

Now compute $f(a + h) - f(a)$ :

$f(a + h) - f(a) = (3a^2 + 6ah + 3h^2 + a + h) - (3a^2 + a) = 6ah + 3h^2 + h$

Thus, substituting back into the limit gives us:

$f'(a) = \lim_{h \to 0} \frac{6ah + 3h^2 + h}{h} = \lim_{h \to 0} (6a + 3h + 1) = 6a + 1$

Step 2

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

99%

104 rated

Answer

To solve the integral, we can use substitution. Let:

$u = 1 + e^x \Rightarrow du = e^x dx$

The limits change accordingly: When $x = 0$ , $u = 2$ . Therefore:

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

Step 3

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

96%

101 rated

Answer

Calculating the definite integral:

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

Evaluating from 0 to 2:

$[3 \sin(2) - 3 \sin(0)] = 3\sin(2) - 0 = 3\sin(2)$

Step 4

How many arrangements of the letters in the word ALGBRAIC are possible?

98%

120 rated

Answer

The word ALGBRAIC consists of 9 letters with the letters A and I repeating once. The number of distinct arrangements is given by:

$\frac{9!}{2!} = \frac{362880}{2} = 181440$

Step 5

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

97%

117 rated

Answer

The vowels A, A, I occupy 4 slots: 2nd, 3rd, 5th, and 8th. The arrangements of the vowels:

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

The remaining 5 letters (L, G, B, R, C) can occupy the remaining positions, giving:

$5! = 120$

Thus, the total arrangements are:

$12 \times 120 = 1440$

Step 6

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

97%

121 rated

Answer

The general term in the binomial expansion of $(x^2 - 1)^9$ is given by:

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

For the term independent of $x$ , we require the power of $x$ to be zero, i.e., $2(9-k) = 0$ . Thus:

$9 - k = 0 \Rightarrow k = 9$

The term is:

$T_9 = \binom{9}{9} (x^2)^0 (-1)^9 = -1$

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 ALGBRAIC are possible?

How many arrangements of the letters in the word ALGBRAIC 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 $(x^2 - 1)^9$.

Join the SSCE students using SimpleStudy...