Questions from General Calculus

Q: (a). Use a CAS to differentiate the function and to

(a). Use a CAS to differentiate the function and to simplify the result. (b). Where does the graph of f have horizontal tangents? (c). Graph f and f' on the same screen. Are the graphs consistent wi...

Q: (a). If is a positive integer, prove that d

(a). If is a positive integer, prove that d/dx (sinnx cos mx) = n sinn-1x cos (n + 1) x (b). Find a formula for the derivative of y = cosnx cos nx that is similar to the one in part (a).

Q: Differentiate the function. A (x) = 12/

Differentiate the function. A (x) = 12/x5

Q: Use the Chain Rule to show that if is measured in degrees

Use the Chain Rule to show that if is measured in degrees, then (This gives one reason for the convention that radian measure is always used when dealing with trigonometric functions in calculus: The...

Q: The graph of the derivative f' of a continuous function f is

The graph of the derivative f' of a continuous function f is shown. (a). On what intervals f is increasing? Decreasing? (b). At what values of x does f have a local maximum? Local minimum? (c). On w...

Q: (a). Write |x| = √x2 and use

(a). Write |x| = √x2 and use the Chain Rule to show that d/dx |x| = x/|x| (b). If f (x) = |sin x|, find f'(x) and sketch the graphs of f and f'. Where is f not differentiable? (c). If g (x) = sin |x|,...

Q: If y = f (u) and u = g (

If y = f (u) and u = g (x), where f and g are twice differentiable functions, show that

Q: Make a careful sketch of the graph of f and below it

Make a careful sketch of the graph of f and below it sketch the graph of f' in the same manner as in Exercises 4–11. Can you guess a formula for f' (x) from its graph? f (x) = ex

Q: Make a careful sketch of the graph of f and below it

Make a careful sketch of the graph of f and below it sketch the graph of f' in the same manner as in Exercises 4–11. Can you guess a formula for f' (x) from its graph? f (x) = ln x

Q: Let f (x) = x2. (a).

Let f (x) = x2. (a). Estimate the values of f'(0), f'(1/2), f'(2), and by using a graphing device to zoom in on the graph of f. (b). Use symmetry to deduce the values of f'(-1/2), f'(-1), and f'(-2)....