I just finished doing my homework on Local Linear Approximations in 3-space (Ch.13.4). In one of the problems the answer I got is different from the answer key.
Problem 39. We have a function $f(x,y,z)=xe^{yz}$ and points $P(1,-1,-1)$ and $Q(0.99,-1.01,-0.99)$
Part a) Find the local Linear Approximation at point $P$
Part b) Compare the error in approximation at point $Q$ with the distance between $P$ and $Q$.( By compare they mean $\frac{\text{error in approximation}}{\text{distance between $P$ and $Q$}}$).
So the ration I get in part b) is $(-0.00982)$, but in the answer key it is $0.01554$
Attempt: Part a) First I found the local linear approximation at point $P$: $L(x,y,z)=e+e(x-1)-e(y+1)-e(z+1)$
Part b)The local linear approximation at $Q$: $L(0.99,-1.01,-0.99)=e+e(-0.01)-e(-0.01)-e(0.01)=2.691$
The actual value of $f(x,y,z)$ at $Q$ is $f(0.99,-1.01,-0.99)=2.6908299$.
The error in approximation: $f(0.99,-1.01,-0.99)-L(0.99,-1.01,-0.99)= -0.0001701$
Distance between points $P$ and $Q$: $D=\sqrt{0.01^2+0.01^2+0.01^2}=0.01732$
So now when I compare the error in app and distance I get: $\frac{\text{error in approximation}}{\text{distance between $P$ and $Q$}}=-0.00982$
But the answer key say it should be $0.01554$. Is it a typo or my mistake?