Questions from General Calculus

Q: Find points P and Q on the parabola y == 1 2x2

Find points P and Q on the parabola y == 1 2x2 so that the triangle ABC formed by the x-axis and the tangent lines at P and Q is an equilateral triangle. (See the figure.)

Q: Find the point where the curves y = x3 - 3x +

Find the point where the curves y = x3 - 3x + 4 and y = 3(x2 – x) are tangent to each other, that is, have a common tangent line. Illustrate by sketching both curves and the common tangent.

Q: Show that the tangent lines to the parabola y = ax2 +

Show that the tangent lines to the parabola y = ax2 + bx + c at any two points with x-coordinates p and q must intersect at a point whose x-coordinate is halfway between p and q.

Q: Show that sin-1(tanh x) = tan-

Show that sin-1(tanh x) = tan-1(sinh x).

Q: Calculate y’. y = ln (x ln x)

Calculate y’. y = ln (x ln x)

Q: Calculate y’. y = emx cos nx

Calculate y’. y = emx cos nx

Q: Calculate y’. y = (arcsin 2x)2

Calculate y’. y = (arcsin 2x)2

Q: Find y’’ by implicit differentiation. sin y + cos x

Find y’’ by implicit differentiation. sin y + cos x = 1

Q: Calculate y’. y = ln sec x

Calculate y’. y = ln sec x

Q: Calculate y’. y + x cos y = x2y

Calculate y’. y + x cos y = x2y