Question

If $f(1)=10,f(2)=14$, then by using Newton’s forward formula $f(1.3)$is equal to

& A.12.2 \\\ & B.11.2 \\\ & C.10.2 \\\ & D.15.2 \\\ \end{aligned}$$

Answer 1

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

& \Rightarrow h=2-1 \\\ & \Rightarrow h=1....(4) \\\ \end{aligned}$$ From the question, it is clear that we should find the value of $$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{aligned} & \Rightarrow m=\dfrac{1.3-1}{1} \\\ & \Rightarrow m=0.3......(7) \\\ \end{aligned}$$ We know that $$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{aligned} & f(1)=10.....(9) \\\ & f(2)=14.....(10) \\\ \end{aligned}$$ Now we will substitute equation (3), equation (4), equation (5), equation (6), equation (7), equation (9) and equation (10) in equation (8), then we get $$\begin{aligned} & \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{aligned}$$ Now from equation (11), it is clear that the value of $$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.