SM 3. Compute the directional derivatives of the following functions at the given points and in the given directions. (a) f(x, y) = 2x + y - 1 at (2, 1), in the direction given by (1, 1). (b) g(x, y, z) = xe – xy -z? at (0, 1, 1), in the direction given by (1, 1, 1).
Angles in Circles
Angles within a circle are feasible to create with the help of different properties of the circle such as radii, tangents, and chords. The radius is the distance from the center of the circle to the circumference of the circle. A tangent is a line made perpendicular to the radius through its endpoint placed on the circle as well as the line drawn at right angles to a tangent across the point of contact when the circle passes through the center of the circle. The chord is a line segment with its endpoints on the circle. A secant line or secant is the infinite extension of the chord.
Arcs in Circles
A circular arc is the arc of a circle formed by two distinct points. It is a section or segment of the circumference of a circle. A straight line passing through the center connecting the two distinct ends of the arc is termed a semi-circular arc.
Can you help with question 3?
![OR SECTION 2.1
1. Compute the gradients of the following functions at the given points.
(a) f(x,y)= y² + xy at (2, 1)
(b) g(x, y,z) = xe*y - z? at (0, 0, 1)
(c) h(x,y,z) = e* + e?Y + e3? at (0, 0, 0)
(d) k(x, y, z) = e*+2y+3% at (0,0,0)
2. Let f(t) be a C' function of t with f'(t) # 0.
(a) Put F(x, y) = f(x² + y?). Find the gradient VF at an arbitrary point and show that it is
parallel to the straight line segment joining the point and the origin.
(b) Put G(x, y)
orthogonal to the straight line segment joining the point and the origin.
f(y/x). Find VG at an arbitrary point where x # 0, and show that it is
SM 3. Compute the directional derivatives of the following functions at the given points and in the
given directions.
(a) f(x, y) = 2x + y - 1 at (2, 1), in the direction given by (1, 1).
(b) g(x, y, z) = xe²Y – xy – z? at (0, 1, 1), in the direction given by (1, 1, 1).
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