Questions from General Calculus

Q: Let P (t) be the percentage of Americans under the

Let P (t) be the percentage of Americans under the age of 18 at time t. The table gives values of this function in census years from 1950 to 2000. (a). What is the meaning of P'(t)? What are its uni...

Q: An elastic band is hung on a hook and a mass is

An elastic band is hung on a hook and a mass is hung on the lower end of the band. When the mass is pulled downward and then released, it vibrates vertically. The equation of motion is s = 2 cos t + 3...

Q: Show that the curve with parametric equations x = sin t,

Show that the curve with parametric equations x = sin t, y = sin (t + sin t) has two tangent lines at the origin and find their equations. Illustrate by graphing the curve and its tangents.

Q: (a). If f (x) = (x2 –

(a). If f (x) = (x2 – 1)/ (x2 + 1), find f'(x) and f"(x). (b). Check to see that your answers to part (a) are reasonable by comparing the graphs of f, f', and f".

Q: The cycloid was x = r (θ – sin θ),

The cycloid was x = r (θ – sin θ), y = r (1 – cos θ) discussed in Example 7 in Section 1.7. (a). Find an equation of the tangent to the cycloid at the point where θ = π/3. (b). At what points is the t...

Q: Computer algebra systems have commands that differentiate functions, but the form

Computer algebra systems have commands that differentiate functions, but the form of the answer may not be convenient and so further commands may be necessary to simplify the answer. (a). Use a CAS to...

Q: An object with weight W is dragged along a horizontal plane by

An object with weight W is dragged along a horizontal plane by a force acting along a rope attached to the object. If the rope makes an angle θ with the plane, then the magnitude of the f...

Q: Graph the function f (x) = x + √|x

Graph the function f (x) = x + √|x|. Zoom in repeatedly, first toward the point (-1, 0) and then toward the origin. What is different about the behavior of f in the vicinity of these two points? What...

Q: Trace or copy the graph of the given function f. (

Trace or copy the graph of the given function f. (Assume that the axes have equal scales.) Then use the method of Example 1 to sketch the graph of f' below it.

Q: Find equations of the tangents to the curve x = 3t2 +

Find equations of the tangents to the curve x = 3t2 + 1, y = 2t3 + 1 that pass through the point (4, 3).