Answer:
❤️
Step-by-step explanation:
(a) If there is a scalar function f(x, y) such that ∇ f(x, y) = x² i + y² j, then
∂f/∂x = x²
∂f/∂y = y²
Integrating both of these equations with respect to x and y (respectively) gives
f(x, y) = 1/3 x³ + g(y)
f(x, y) = 1/3 y³ + h(x)
Differentiating with respect to the other variable gives
∂f/∂y = g'(y) = y²
∂f/∂x = h'(x) = x²
so it follows that
f(x, y) = 1/3 x³ + 1/3 y³ + C
for some constant C.
(b) By the gradient theorem,

So 3 times the measure of a small angle
3x + x= 180
4x = 180
X= 45
3(45) + 4(45) = 180
Hope this helps!
Answer: b) slope: -1, y-intercept 12
Step-by-step explanation:
The equation here is in slope-intercept form (y=mx+b)
In slope-intercept form m is the slope and b is the y-intercept
m=-1 and b=12 so the answer is b
Answer:
Part 19) x=5
Part 20) x=-3
Step-by-step explanation:
Problem 19) Find the value of x so that f(x)=7
step 1
Find the intersection point of the given line in the graph with the line y=7
The intersection point is (5,7)
see the attached figure
therefore
The x-coordinate of the intersection point is the value of x when f(x)=7
x=5
f(5)=7
Problem 20) Find the value of x so that f(x)=7
step 1
Find the intersection point of the given line in the graph with the line y=7
The intersection point is (-3,7)
see the attached figure
therefore
The x-coordinate of the intersection point is the value of x when f(x)=7
x=-3
f(-3)=7