Finding Antiderivatives
In Exercises 1–16, find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation.
x⁷

Checking Antiderivative Formulas

Right, or wrong? Say which for each formula and give a brief reason for each answer.

∫xsinx dx = -x cos x + C

[Technology Exercise] 16. Designing a box with a lid A piece of cardboard measures 10 in. by 15 in. Two equal squares are removed from the corners of a 10-in. side as shown in the figure. Two equal rectangles are removed from the other corners so that the tabs can be folded to form a rectangular box with lid.

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b. Find the domain of V for the problem situation and graph V over this domain.

114. Parabolas

b. When is the parabola concave up? Concave down?

[Technology Exercise] 17. Designing a suitcase A 24-in.-by-36-in. sheet of cardboard is folded in half to form a 24-in.-by-18-in. rectangle as shown in the accompanying figure. Then four congruent squares of side length x are cut from the corners of the folded rectangle. The sheet is unfolded, and the six tabs are folded up to form a box with sides and a lid.

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b. Find the domain of V for the problem situation and graph V over this domain.

38. What values of a and b make f(x) = x^3 + ax^2 + bx have

b. a local minimum at x = 4 and a point of inflection at x = 1?

Theory and Examples

Cubic functions Consider the cubic function f(x) = ax³ + bx² + cx + d.

b. How many local extreme values can f have?