University:
California State University, Northridge
Course:
MATH 250 | Calculus III
Academic year:

2024
Views:

92

Pages:

2
Author:

Mia Hobbs

Question 1 Show that the curve y = root of(sin x) and y = root of(cos x) are intersecting perpendicularly at 0 < x < phi/2. Answer 039;s start by finding the derivatives of the given curves. For y = root of(sin x), we can use the chain rule to get: dy/dx = (1/2)(sin x)^(-1/2) * cos x Similarly, for y = root of(cos x), we get: dy/dx = (1/2)(cos x)^(-1/2) * (-sin x) Next, we need to find the points of intersection between the two curves. Setting y = root of(sin x) equal to y = root of(cos x), we get: root of(sin x) = root of(cos x) Simplifying, we get: sin x = cos x Using trigonometric identities, we know that sin x = cos x at x = pi/4. However, since we are only considering values at 0 < x < phi/2, we can narrow it down to x = phi/4. Now, we can apply the First Derivative Test to these points of intersection. Plugging in x = phi/4 into the derivatives we found earlier, we get: dy/dx = (1/2)(sin phi/4)^(-1/2) * cos phi/4 = 2 and dy/dx = (1/2)(cos phi/4)^(-1/2) * (-sin phi/4) = -2 Since the first derivative at x = phi/4 is positive for y = root of(sin x) and negative for y = root of(cos x), we can conclude that the curves intersect perpendicularly at this point. Furthermore, since the first derivative is changing from positive to negative (or vice versa), we can also conclude that the curves intersect at a right angle (perpendicular) at this point. Question 2 Consider tha graph to the function f(x)=2k^3x^3+3kx^2+12x-55, where k is a positive constant. use calculus to find expressions, in terms of k, for the range(s) of values of x for which this graph is increasing. Justify your choice of range(s) clearly Answer 039;(x) is negative to the left and positive to the right, then the critical point is a maximum value. Using this test, we can evaluate f& Question 3 Consider tha graph to the function f(x)=2k^3x^3+3kx^2+12x-55, where k is a positive constant. use calculus to find expressions, in terms of k, for the range(s) of values of x for which this graph is increasing. Justify your choice of range(s) clearly Answer 039;(x) at both critical points, -√(2k/3) and √(2k/3), to determine if they represent a minimum or maximum value. For k within the range of 0≤k≤8, both critical points result in f&

Calculus Answer Note #2

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