Questions from General Calculus

Q: The altitude of a triangle is increasing at a rate of 1

The altitude of a triangle is increasing at a rate of 1 cm/min while the area of the triangle is increasing at a rate of 2 cm2/min. At what rate is the base of the triangle changing when the altitude...

Q: Show that the curves y = e-x and y =

Show that the curves y = e-x and y = -e-x touch the curve y = e-x sin x at its inflection points.

Q: Rainbows are created when raindrops scatter sunlight. They have fascinated mankind

Rainbows are created when raindrops scatter sunlight. They have fascinated mankind since ancient times and have inspired attempts at scientific explanation since the time of Aristotle. In this project...

Q: Show that a cubic function (a third-degree polynomial)

Show that a cubic function (a third-degree polynomial) always has exactly one point of inflection. If its graph has three x-intercepts x1, x2 and x3, show that the x-coordinate of the inflection point...

Q: (a). Find the intervals on which f is increasing or

(a). Find the intervals on which f is increasing or decreasing. (b). Find the local maximum and minimum values of f. (c). Find the intervals of concavity and the inflection points. f (x) = 2x3 + 3x2 -...

Q: Sketch the graph of f by hand and use your sketch to

Sketch the graph of f by hand and use your sketch to find the absolute and local maximum and minimum values of f. (Use the graphs and transformations of Sections 1.2 and 1.3.) f (x) = 1 - √x

Q: (a). Use a graph of f to give a rough

(a). Use a graph of f to give a rough estimate of the intervals of concavity and the coordinates of the points of inflection. (b). Use a graph of f" to give better estimates. f (x) = x3(x – 2)4

Q: Find the critical numbers of the function. f (x

Find the critical numbers of the function. f (x) = x3 + x2 + x

Q: For what values of does the polynomial P (x) =

For what values of does the polynomial P (x) = x4 + cx3 + x2 have two inflection points? One inflection point? None? Illustrate by graphing P for several values of c. How does the graph change as decr...

Q: (a). If the function f (x) = x3

(a). If the function f (x) = x3 + ax2 + bx has the local minimum value -2/9√3 at 1/√3, what are the values of a and b? (b). Which of the tangent lines to the curve in part (a) has the smallest slope?...