Suppose that g(z) = 4z^2 + 7z. a.) Compute and simplify...

Suppose that $g(z) = 4z^2 + 7z$. a.) Compute and simplify $g(z+h) - g(z)$. $g(z+h) - g(z) = 8zh + 4h^2 + 7h$ b.) Find a formula for $g'(z)$. $g'(z) = 8z + 7$ c.) Compute $g(1)$ and $g'(1)$. $g(1) = 11$ $g'(1) = 15$ d.) Use the approximation $g'(1) \approx \frac{g(1.0003) - g(1)}{0.0003}$ and the values you found in part (c) to estimate the value of $g(1.0003)$ without plugging 1.0003 into the formula for $g(z)$. $g(1.0003) \approx 15.0012$

Transcript

00:01 So we have some matrices given like that is z cube plus b z square plus c z is equal to d so that means to z cube z cube z cube to z cube plus minus 3 z square 0 minus 2 z square minus 5 z square plus minus 2 z square plus minus 2 z z squared minus 4 z 0 to z is equal to d which is 0 000 adding this elements and equating to 0 you have 2 z cube minus 3 z square minus 2 z is equal to 0 or z into 2 z square minus 3 z minus 2 is equal to 0 because when you add matrices corresponding elements also gets added and my two matrices are equal corresponding element also equal.

01:07 Solving this, we get z equal to 0 and you can factorize this, you can let this as to z squared minus z plus z minus 2 is equal to 0, so that means z is equal to 0 or 2 or negative 1, uh, negative half, sorry, negative half.

01:34 But also z cube plus 0 minus 4c is 0.

01:39 Cube plus 4 z i mean to 0 z q plus 0 minus 4 z is equal to 0 so that means z cube is equal to 4c z is equal to 0 2 or negative 2 this z cube minus 2 z q minus 2 z squared is equal to 0 means z is double root of 0 0 0 or 2 next last 1 2 2 2 2 q q minus 5 5 x squared plus 2 z is 0 2 z cube minus 5 z squared plus 2 z squared plus 2 z is equal to 0 so that means z z into 2 z square minus 5 z plus 2 is equal to 0 and this will give you 2 z to 2 z minus 1 into z minus 2 is equal to 0 that means 0 a half or 2 so in all these which is the common values of z so this one we got for this and for this this this this you can see zero is common two is common minus half is not common because minus half is absent in this and this minus two is not common because minus two is not there in this and this so the values of z satisfy this equation is z q plus b z squared to c z plus a b c d if a b c d are these matrices this is zero and two.

03:16 Let us go to the next set.

03:20 Doing the same way, let's directly write 1 into z cube, 1 into z cube minus 3 into z squared plus z is equal to 5 into negative 1.

03:33 This is one equation next to 1 into z cube minus 5 into z square plus 9c is equal to 5...

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