Question

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

Answer 1

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 \[f(1.3)\].

Complete step by step answer:
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.