Questions from General Calculus

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) = x4e-x

Q: Sketch the graph of a function that satisfies all of the given

Sketch the graph of a function that satisfies all of the given conditions. Vertical asymptote x = 0, f ‘(x) > 0 if x < -2, f ‘(x) < 0 if x > -2 (x ≠ 0), f ‘‘(x) < 0 if x < 0, f ‘‘(x) > 0 if x > 0

Q: Two curves are orthogonal if their tangent lines are perpendicular at each

Two curves are orthogonal if their tangent lines are perpendicular at each point of intersection. Show that the given families of curves are orthogonal trajectories of each other; that is, every curve...

Q: Sketch the graph of a function that satisfies all of the given

Sketch the graph of a function that satisfies all of the given conditions. f ‘(0) = f ‘(2) = f ‘(4) = 0, f ‘(x) > 0 if x > 0 or 2 < x < 4, f ‘(x) < 0 if 0 < x < 2 or x > 4, f ‘‘(x) > 0 if 1 < x < 3,...

Q: Show that the ellipse x2/a2 + y2/b2 =

Show that the ellipse x2/a2 + y2/b2 = 1 and the hyperbola x2/A2 - y2/B2 = 1 are orthogonal trajectories if A2, a2 and a2 - b2 = A2 + B2 (so the ellipse and hyperbola have the same foci).

Q: Sketch the graph of a function that satisfies all of the given

Sketch the graph of a function that satisfies all of the given conditions. f ‘(x) > 0 for all x ≠ 1, vertical asymptote x = 1, f ’’(x) > 0 if x < 1 or x > 3, f ’’(x) < 0 if 1 < x < 3

Q: Find the value of the number a such that the families of

Find the value of the number a such that the families of curves y = s(x + c)-1 and y = a(x + k)1/3 are orthogonal trajectories.

Q: (a) Use implicit differentiation to find y’ if x2 +

(a) Use implicit differentiation to find y’ if x2 + xy + y2 - 1 = 0 (b) Plot the curve in part (a). What do you see? Prove that what you see is correct. (c) In view of part (b), what can you say about...

Q: The equation x2 - xy + y2 = 3 represents a “

The equation x2 - xy + y2 = 3 represents a “rotated ellipse,” that is, an ellipse whose axes are not parallel to the coordinate axes. Find the points at which this ellipse crosses the x-axis and show...

Q: Illustrate l’Hospital’s Rule by graphing both f (x)/g(

Illustrate l’Hospital’s Rule by graphing both f (x)/g(x) and f ’(x)/g’ (x) near x = 0 to see that these ratios have the same limit as x ( 0. Also, calculate the exact value of the limit. f (x) = ex -...