Questions in Section Calculus I: page 5

Find the expression for `sin(cos^(-1)(x))` and `tan(cos^(-1)(x))`.

Find the natural domain of the following functions. Explain your results.

  1. `f(x)=x/([x])`.
  2. `h(x) = 1 - sin(x)`.

The circumference of a sphere was measured to be 84 cm with a possible error of 0.5 cm.

  1. Use differentials to estimate the maximum error in the calculated surface area. Find the relative error.
  2. Use differentials to estimate maximum percentage error in the calculated volume.
  1. Find the linearization of `f(x)=e^x` at `a=0`.
  2. Using a) find an approximate value of `e^(-0.012)`.

A particle is moving along a hyperbola `xy=8`. The vertical velocity (i.e. the rate at which `y` is decreasing) has magnitude 3 cm per minute, when it reaches `(4,2)`. What is the horizontal velocity at `(4,2)`?

You wish to form a fence with the shape of a rectangle and a semi-circle joined together. Rounded to the nearest whole number, what is the minimum length of fence needed such that area is 200 square yards?

At how many points do the tangents to the functions `y=3^x` and `y=x^3` have the same slope?

In the quadrilateral PQRS the side QR has length 1 meter, the side RS has length 2 meters, and the angle at R is a right angle. The diagonal PR has length 2 meters. The point P is a perpendicular distance `x` meters from QR. The value of `x` is between 0 and 2. (The quadrilateral described cannot exist for other values of `x`.)The total area `A(x)` of the quadrilateral is given by `A(x)=1/2x+sqrt(4-x^2)` (`0<x<2`).

Use implicit differentiation to find `(dy)/(dx)` given `(x-2)^2+y^3=2xy^2`. Find an equation for the tangent line at point `(1,1)`.

  1. Given `f(x)=log_10(sin(3x))`, find derivative `f'(x)`.
  2. Given that `f(x)=(e^(2x))/(3x-1)` find derivative `f'(x)` and `f'(1)`.
  3. Use the first principle to find derivative `d/(dx)x^3`.
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