Question

# If $f(1)=10,f(2)=14$, then by using Newton’s forward formula $f(1.3)$is equal to\begin{align} & A.12.2 \\ & B.11.2 \\ & C.10.2 \\ & D.15.2 \\ \end{align}

Hint: We should know that the formula of newton’s forward method is used as follows: If$f(x)=f(a)+m\left( f(b)-f(a) \right)$ where a and b are two integers, $a < x < b$, $m=\dfrac{x-a}{h}$ where $h=b-a$. By using this formula, we can find the value of $f(1.3)$.

Before solving the question, we should know how the newton’s forward formula is used. If$f(x)=f(a)+m\left( f(b)-f(a) \right)$ where a and b are two integers, $a\le x\le b$, $m=\dfrac{x-a}{h}$ where $h=b-a$.
From the question, it was given that $f(1)=10,f(2)=14$. Now we should find the value of $f(1.3)$ using newton’s forward formula.
Now let us compare $f(1)=10,f(2)=14$ with $f(a),f(b)$. Then the value of a is equal to 1 and the value of b is equal to 2.
We know that $h=b-a$.
Let us consider
$h=b-a....(3)$
Now let us substitute equation (1) and equation (2) in equation (3), then we get
\begin{align} & \Rightarrow h=2-1 \\ & \Rightarrow h=1....(4) \\ \end{align}
From the question, it is clear that we should find the value of $f(1.3)$.
Now we will let us assume $f(x)$ with $f(1.3)$.
Then, we get
$\Rightarrow x=1.3....(5)$
We know that $m=\dfrac{x-a}{h}$.
Let us consider
$m=\dfrac{x-a}{h}......(6)$
So, let us substitute equation (5), equation (1) and equation (4) in equation (6), then we get
\begin{align} & \Rightarrow m=\dfrac{1.3-1}{1} \\ & \Rightarrow m=0.3......(7) \\ \end{align}
We know that $f(x)=f(a)+m\left( f(b)-f(a) \right)$ where a and b are two integers, $a\le x\le b$, $m=\dfrac{x-a}{h}$ where $h=b-a$.
Let us consider
$f(x)=f(a)+m\left( f(b)-f(a) \right).....(8)$
We know that $f(1)=10,f(2)=14$.
So, let us consider
\begin{align} & f(1)=10.....(9) \\ & f(2)=14.....(10) \\ \end{align}
Now we will substitute equation (3), equation (4), equation (5), equation (6), equation (7), equation (9) and equation (10) in equation (8), then we get
\begin{align} & \Rightarrow f(1.3)=10+\left( 0.3 \right)\left( 14-10 \right) \\ & \Rightarrow f(1.3)=10+\left( 0.3 \right)\left( 4 \right) \\ & \Rightarrow f(1.3)=10+1.2 \\ & \Rightarrow f(1.3)=11.2......(11) \\ \end{align}
Now from equation (11), it is clear that the value of $f(1.3)$ is equal to 11.2.

So, the correct answer is “Option B”.

Note: Students may have a misconception that If$f(x)=f(a)+m\left( f(a)-f(b) \right)$ where a and b are two integers, $a < x < b$, $m=\dfrac{x-a}{h}$ where $h=b-a$. If this misconception is followed then the whole problem may go wrong. So, this misconception should be avoided. Students should not make any calculation mistakes while solving the problem to get a correct solution.