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# MTH 3030: Final Practice 1

Problem 1

Determine whether the series $$\displaystyle \sum_{n=1}^{\infty}\frac{n^{6}-7n^{5}+8}{4n^{7}+7n^{6}+4}$$ is convergent or divergent.

Problem 2

Find a symmetric equation of the plane that contains the points $$A(5,3,6)$$, $$B(0,3,9)$$, and $$C(4,-6,5)$$.

Problem 3

Assume that $$A(t)=t^3-2t$$, $$B(t)=2t+5$$, and $$C(t)=t^2+5t$$, and assume that $$f(x,y,z)$$ is a differentiable function such that $$\nabla f(-1,7,6)=\langle 3,6,4\rangle$$. Assume that the function $$g$$ is defined as $$g(t)=f(A(t),B(t),C(t))$$. Determine $$g^{\prime}(1)$$.

Problem 4

Determine whether the series $$\displaystyle \sum_{n=1}^{\infty}\frac{7-3\sin^3 n}{n+6}$$ is convergent or divergent.

Problem 5

Evaluate the integral $$\displaystyle\int_{1}^{3}\int_{2}^{5} \int_{3}^{5}\left(3z-y\right)\,dydzdx$$.

Problem 6

Consider the region $$D$$ in the $$xy$$ plane defined as $$D=\left\{(x,y): 10\leq 10x \leq y\leq x^2+25\right\}$$.

• (a) Sketch the region $$D$$.

• (b) Evaluate the integral $$\displaystyle \iint_Dx\,dA$$.

Problem 7

Determine and classify the critical points of the function $$f(x,y)=195y+18xy+y^3-3x^2+30$$.

Problem 8

Evaluate the integral $$\displaystyle\iint_R \frac1{x^2y}\,dA$$ where $$R$$ is the set of points $$(x,y)$$ that satisfy the following two conditions: \begin{eqnarray*} 3x+7\leq &y&\leq 3x+9\newline 4x\leq &y&\leq 8x. \end{eqnarray*}

Problem 9

The minimum of the differentiable function $$F(x,y)$$ on the curve $$x^2=6y^2-15$$ is attained at the point $$(3,2)$$. It is given that $$F_x(3,2)=24$$. Determine $$F_y(3,2)$$.

Problem 10

For each of the following statements determine whether it is true or false. If the statement is true, provide a mathematically rigorous proof of the statement. If the statement is false, provide a counter-example.

• (a) If $$(a_n)$$ is a sequence of positive real numbers such that $$\displaystyle\frac{a_n}2\leq a_{n+1}\leq \frac{a_n+\frac{a_n}n}2$$ then the series $$\displaystyle \sum_{n=1}^{\infty}a_n$$ is convergent.

• (b) If $$f$$ is a positive function such that $$\displaystyle \lim_{x\to+\infty} f(x)=0$$ and the integral $$\displaystyle \int_1^{+\infty}f(x)\,dx$$ is convergent, then the series $$\displaystyle \sum_{n=1}^{\infty}(-1)^nf(n)$$ is convergent.

• (c) Assume that $$(a_n)$$ and $$(b_n)$$ are sequences of real numbers such that $$\displaystyle \sum_{n=1}^{\infty} a_n$$ and $$\displaystyle \sum_{n=1}^{\infty} b_n$$ are convergent. Then the series $$\displaystyle \sum_{n=1}^{\infty} a_nb_n$$ is convergent as well.

• (d) Assume that $$(a_n)$$ and $$(b_n)$$ are sequences of real numbers such that $$a_n\geq 0$$ and $$b_n\geq 0$$ for all $$n$$. If $$\displaystyle \sum_{n=1}^{\infty}(-1)^na_n$$ and $$\displaystyle \sum_{n=1}^{\infty}(-1)^nb_n$$ are convergent, then $$\displaystyle \sum_{n=1}^{\infty}(-1)^na_nb_n$$ is convergent as well.