Q. 2 - Calculus

Question

In each problem that follows we are given values for $f\left(c\right)$and $f\text{'}\left(c\right)$ for some $c$ Use this information to approximate $f(c+h)$ for the given value of $h$.

a)$f\left(0\right)=2,f\text{'}\left(0\right)=1,h=1$

b)$f\left(1\right)=-1,f\text{'}\left(1\right)=-3,h=0.5$

c)$f(-2)=-3,f\text{'}(-2)=2,h=0.25$

d)$f(1.5) = 6.2, f\text{'} (1.5) = 2.2, h = 0.1$

Step-by-Step Solution

Answer

The linear approximation are:

a)$f\left(1\right)=3$

b)$f(1.5)=-2.5$

c)$f(-1.75)=-2.5$

d)$f(1.6)=6.42$

Step 1. Given information

a)$f\left(0\right) = 2, f\text{'}\left(0\right) = 1, h = 1$

b)$f\left(1\right) = -1, f\text{'}\left(1\right) = -3, h = 0.5$

c)$f(-2) = -3, f\text{'} (-2) = 2, h = 0.25$

d)$f(1.5) = 6.2, f\text{'} (1.5) = 2.2, h = 0.1$

Part(a) step 1. Evaluate linear approximation.

The linear approximation is given by a formula:$f(c+h)=f\left(c\right)+hf\text{'}\left(c\right)$

Given $$\begin{gathered} f(0+1)=f\left(0\right)+1.f\text{'}\left(0\right) \\ f\left(1\right)=2+1\left(1\right) \\ f\left(1\right)=3 \end{gathered}$$

Part (b) step 1. Evaluate linear approximation.

$$\begin{gathered} f(c+h)=f\left(c\right)+hf\text{'}\left(c\right) \\ f(1+0.5)=f\left(1\right)+(0.5)f\text{'}\left(1\right) \\ f(1.5)=-1+(0.5)(-3) \\ f(1.5)=-1-1.5 \\ f(1.5)=-2.5 \end{gathered}$$

Part(c) step 1. Evaluate linear approximation.

$$\begin{gathered} f(c+h)=f\left(c\right)+hf\text{'}\left(c\right) \\ f(-2+0.25)=f(-2)+(0.25)f\text{'}(-2) \\ f(-1.75)=-3+(0.25)\left(2\right) \\ f(-1.75)=-3+0.5 \\ f(-1.75)=-2.5 \end{gathered}$$

Part(d) step 1. Evaluate linear approximation.

$$\begin{gathered} f(c+h)=f\left(c\right)+hf\text{'}\left(c\right) \\ f(1.5+0.1)=f(1.5)+(0.1)f\text{'}(1.5) \\ f(1.6)=6.2+(0.1)(2.2) \\ f(1.6)=6.2+0.22 \\ f(1.6)=6.42 \end{gathered}$$