1. Find the equation of the tangent line to: a) y = x2 – 3 at the point (2,1) b) y = cos x at the point (1,1) c) y=e" at the point where r = 1 d) r3 + y3 = 19 at the point (3,-2) 2. Find the equation of the normal line to: a) y = r at the point (2,8) b) y=x+ at the point where x = 2 c) y = 2:03 - 5x +...
let F(x,y) = 3x^2y^2i+2x^3yj and c be the path consisting of
line segments from(1,2) to (-1,3), from (-1,3) to (-1,1), and from
(-1,1) to (2,1). evaluate the line integral of F along c.
Let F(x, y) = 3x²y2 i + 2x’yj and C be the path consisting of line segments from (1, 2) to (-1,3), from (-1, 3) to (-1, 1), and from (-1, 1) to (2, 1). Evaluate the line integral of F along C.
4. Let f(x, y) = 2 - 2x – y + xy. (a) Find the directional derivative of f at the point (2,1) in the direction (-1,1). [2] (b) Find all the critical points of the function f and classify them as local extrema, saddle points, etc. [2]
Find the point at which the line fix)--2x-3 intersects the line g(x) -x (x, y) =(1-3,-x Recall that whhen two lines intersect, their point of intersection has the same y-coordinate as well as the same x-coordinate. Thus, if rx)-y and g(x)-y, then fx)-g(x). What are the two expressions in terms of x that are set equal to each other and then solved for x?
3. Find m for which the following lines do not form a triangle. x+2y 5D, 2x-3y-4 .2, mx+y 0 [Sol] Since line 3 passes through the origin, lines D, 2 and 3will not form a triangle in the following three cases: wor When D and 3 are (i) parallel. doa When 2 and 3 are (ii) parallel. When D, and 3 all intersect at one point. (ii Therefore, from ( i ), (ii) and (iii), m 4. Find a for...
Three lines are defined by the three equations: x + y = 0 x-y=0 2x+y=1 The three lines form a triangle with vertices at: O A. (0,0),(33), 1,-1) O B. (0,0), (17 2 ) (-1,-1) o C. (1,1), (1, -1), (2, 1) O D. (1,1),(3,-3), (-2,-1) THE CORRECT ANSWER IS: A y=-x y=x -2x + 1 y=-2x +1 intersection of y=x and y=-x is (0,0) intersection of y = x and y = -2x + 1 is x = -2x...
2. Find the equation of a line which is perpendicular to the line 2x-y=4 and include P(3,4) 3. Find the equation of a cirle with the radius 3 and the center (1,-1) 2.5 4. Find the radius and the center of the following cicle equation 05 x² + y² - 2x + 4y - 4 = 0 1. GÖ
When the line y = 2x + 3 is reflected in the line y = x + 1, what is the equation of the reflected line?
The variables x and y are implicitly related to the equation x^4+ { ^Y down 1 e^-t^2 dt =1 ( Y is at the top of the { and 1 is at the bottom of the { ) The point p=(1,1) lies on the graph of the equation. Find the slope of the line tangent to the graph at the point p=(1,1) A.) 2e^-2 B.) 2e C.) -4e D.) -4e^-1 E.) 4e^-2
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2 t, 3- 3 and (-2r, 3-, ) 11.) Determine the point of intersection of the lines Note:4,1,2, with (1,2/3,-1)^k(-2.-1,1)=kv;, for any kER. So .1,. Change one of the parameters to s, then equate the corresponding coordinates of the lines and solve for t, and s. substitute the values of t, and s in their respective lines to get the required point. Locate the point of intersection of the plane 2x+ y-z-0 and the line through (3,1,0)...