Let's break down the questions in the image.

Question 4

1. Given Function: $y = \sqrt{x - 5}$.
2. Task: Use the first principles (the definition of the derivative) to find $\frac{dy}{dx}$.

Using the first principles, the derivative of $y$ with respect to $x$ can be defined as: $\frac{dy}{dx} = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$ where $f(x) = \sqrt{x - 5}$.

Substitute $f(x + h) = \sqrt{x + h - 5}$ and proceed to simplify using the limit definition.

Question 5

(i) Task: Show that the equation of the normal to the curve is $x + 5y = 33$.

To find the normal, you first need the derivative of the curve at the point of interest, then find the negative reciprocal to get the slope of the normal.

(ii) Task: Find the coordinates where the normal intersects the x-axis or y-axis.

Would you like detailed steps for each part?

Related Questions

1. What are first principles in calculus, and why are they used for derivatives?
2. How do you simplify expressions when using first principles with square roots?
3. How do you determine the equation of a tangent line?
4. What is the difference between the tangent and normal lines to a curve?
5. How do you find intersection points of a line with the axes?

Tip: For functions involving square roots, rationalizing the expression helps in simplifying limits when using first principles.