Use Definition 5 and the fact that
12+22+32+⋯+n2=16n(n+1)(2n+1)1^2+2^2+3^2+\cdots+n^2=\frac{1}{6} n(n+1)(2 n+1) 12+22+32+⋯+n2=61n(n+1)(2n+1)
to show that the area described in Example 2 is exactly 13\frac{1}{3}31.
Sketch the slope field for y′+y=2y^{\prime}+y=2y′+y=2 at the 25 gridpoints (x,y)(x, y)(x,y), where x=0,1,…,4x=0,1, \ldots, 4x=0,1,…,4 and y=0,1,…,4y=0,1, \ldots, 4y=0,1,…,4.
Solve for t in each of the following compound interest equations. Leave your answer in terms of a logarithm. 5490=4800(1.009)3t5490=4800(1.009)^{3 t}5490=4800(1.009)3t
Suppose f is differentiable on R\mathbb{R}R and α\alphaα is a real number. Let F(x) = f(xα)f\left(x^{\alpha}\right)f(xα) and G(x) = [f(x)]α^{\alpha}α. Find expressions for (a) F′^{\prime}′(x) and (b) G′^{\prime}′(x).