(a) (i) Use differentiation from first principles to show that $$\frac{d}{dx}(x) = 1.$$ (ii) Use mathematical induction and the product rule for differentiation to prove that $$\frac{d}{dx}(x^n) = nx^{n-1}$$ for all positive integers $n$. (b) A billboard of height $h$ metres is mounted on the side of a building, with its bottom edge $h$ metres above street level. The billboard subtends an angle $\theta$ at point $P$, $x$ metres from the building. (i) Use the identity: $$\tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B}$$ to show that $$\theta = \tan^{-1}\left[ \frac{\frac{a}{x}}{x^2 + h(l + h)} \right].$$ (ii) The maximum value of $\theta$ occurs when $ \frac{d\theta}{dx} = 0 $ and $x$ is positive. Find the value of $x$ for which $\theta$ is a maximum. (c) Consider the billboard in part (b). There is a unique circle that passes through the top and bottom of the billboard (points $Q$ and $R$ respectively) and is tangent to the street at $T$. Let $\phi$ be the angle subtended by the billboard at $S$, the point where $PQ$ intersects the circle. (i) Show that $\theta < \phi$ when $P$ and $T$ are different points, and hence show that $\theta$ is a maximum when $P$ and $T$ are the same point. (ii) Using circle properties, find the distance of $T$ from the building.

SSCE HSC Mathematics Extension 1 Division of polynomials and the remainder theorem: (a) (i) Use differentiation fr

(a) (i) Use differentiation from first principles to show that $$\frac{d}{dx}(x) = 1.$$ (ii) Use mathematical induction and the product rule for differentiation to prove that $$\frac{d}{dx}(x^n) = nx^{n-1}$$ for all positive integers $n$ - HSC - SSCE Mathematics Extension 1 - Question 7 - 2009 - Paper 1

Question 7

$(a)--(i)-Use-differentiation-from-first-principles-to-show-that--$$\frac{d}{dx}(x)-=-1.$$--(ii)-Use-mathematical-induction-and-the-product-rule-for-differentiation-to-prove-that--$$\frac{d}{dx}(x^n)-=-nx^{n-1}$$-for-all-positive-integers-$n$-HSC-SSCE Mathematics Extension 1-Question 7-2009-Paper 1.png$

(a) (i) Use differentiation from first principles to show that $$\frac{d}{dx}(x) = 1.$$ (ii) Use mathematical induction and the product rule for differentiation ... show full transcript

Worked Solution & Example Answer:(a) (i) Use differentiation from first principles to show that $$\frac{d}{dx}(x) = 1.$$ (ii) Use mathematical induction and the product rule for differentiation to prove that $$\frac{d}{dx}(x^n) = nx^{n-1}$$ for all positive integers $n$ - HSC - SSCE Mathematics Extension 1 - Question 7 - 2009 - Paper 1

Step 1

Show that $$\frac{d}{dx}(x) = 1$$

96%

114 rated

Answer

To prove that ( \frac{d}{dx}(x) = 1 ) using first principles, we apply the definition of the derivative:

$\frac{d}{dx}(f(x)) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$

For our function, this becomes:

$\frac{d}{dx}(x) = \lim_{h \to 0} \frac{(x+h) - x}{h} = \lim_{h \to 0} \frac{h}{h} = \lim_{h \to 0} 1 = 1.$

Step 2

Use mathematical induction to prove that $$\frac{d}{dx}(x^n) = nx^{n-1}$$

99%

104 rated

Answer

To prove ( \frac{d}{dx}(x^n) = nx^{n-1} ) by mathematical induction, we perform the following steps:

Base Case: For ( n = 1 ):

$\frac{d}{dx}(x^1) = \frac{d}{dx}(x) = 1 = 1\cdot x^{1-1}.$

Inductive Step: Assume true for ( n = k ):

$\frac{d}{dx}(x^k) = kx^{k-1}.$

Now consider ( n = k + 1 ):

Using the product rule:

$\frac{d}{dx}(x^{k+1}) = \frac{d}{dx}(x^k \cdot x) = x^k \cdot \frac{d}{dx}(x) + x \cdot \frac{d}{dx}(x^k) = x^k \cdot 1 + x \cdot kx^{k-1} = x^k + kx^k = (k+1)x^k.$

Thus, by induction, the statement holds for all positive integers ( n ).

Step 3

Show that $$\theta = \tan^{-1}\left[ \frac{\frac{a}{x}}{x^2 + h(l + h)} \right]$$

96%

101 rated

Answer

We start with the identity given in the problem:

$\tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B}.$

Letting ( A = \tan^{-1}\left(\frac{a}{h}\right) ) and ( B = \tan^{-1}\left(\frac{h}{x}\right), ) we have:

$\theta = \tan^{-1}\left(\frac{\frac{a}{h} - \frac{h}{x}}{1 + \frac{a}{h}\cdot\frac{h}{x}}\right).$

After simplifying the right-hand side, we can express ( \theta ) as shown in the question.

Step 4

Find the value of $ x $ for which $ \theta $ is a maximum

98%

120 rated

Answer

To find the maximum value of ( \theta ), we take the derivative of ( \theta ) with respect to ( x ) and set it to zero:

$\frac{d\theta}{dx} = 0.$

Solving this equation while keeping in mind the constraints that $x$ is positive will give us the required value of $x$ .

Step 5

Show that $ \theta < \phi $ when P and T are different points

97%

117 rated

Answer

By considering the geometry of the situation, we establish that if points P and T are different, the angle subtended by the billboard will always be less than the angle subtended directly at T when they coincide. This can be mathematically expressed using geometric properties and can be proven via triangle inequalities.

Step 6

Find the distance of T from the building

97%

121 rated

Answer

Using circle properties, we can apply the tangent-secant theorem or similar triangles to establish a relationship between the circles involved. By constructing the triangle that includes the points P, T, and the base of the billboard, we can find the distance by applying the appropriate geometric relationships established within the problem.

Show that $$\frac{d}{dx}(x) = 1$$

Use mathematical induction to prove that $$\frac{d}{dx}(x^n) = nx^{n-1}$$

Show that $$\theta = \tan^{-1}\left[ \frac{\frac{a}{x}}{x^2 + h(l + h)} \right]$$

Find the value of \( x \) for which \( \theta \) is a maximum

Show that \( \theta < \phi \) when P and T are different points

Find the distance of T from the building

Join the SSCE students using SimpleStudy...