Chapter 4.2, Problem 47E

Find the equation of the tangent line to the curve y = 6 sec z – 12 cos a at the point (, 6). equation of this tangent line can be written in the form y = mx + b where m is: The and where b is:

5) Determine the slope of the tangent line to the function with the given information: a. f(x, y) = sin(xy) in the direction 2î - 3ĵ, at the point (, -1). b. f(x, y) = 3+x² y² going directly away from the origin, at the point (4,−1). c. f(x, y) = -6x + 2y + 3 in the direction 30° counter-clockwise from the negative y-axis, at the point (-2,2). d. f(x, y) = x³y² + √3x − y in the direction toward (0,5), at the point (3, 1).

Consider the curve defined by r(t) = (cos(t) + t sin(t))i + (sin(t) - t cos(t))j for t > 0. (a) Compute the derivative (t) of the curve above. Simplify your answer. (b) Use the result above to compute (t). The Pythagorean identity cos² (t) + sin² (t) = 1 will be helpful here. Also, we are assuming t> 0 for convenience. ㅠ (c) Use the result above to find the arc length of F(t) for ≤ t ≤ ñ.