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# MTH 2610: Final Practice 1

Problem 1

Evaluate the limit $$\displaystyle \lim_{x\to 1}\frac{x-1}{\sqrt{ 7x+9 }-4}$$.

Problem 2

Find all numbers $$x$$ that satisfy the equation $$\log_{2}\left(x^2-3x-54\right)=4$$.

Problem 3

Find the area bounded by $$y=48x-12x^2$$ and the $$x$$ axis.

Problem 4

Use the linearization of the function that satisfies the criteria $$f(4)=12$$ and $$f^{\prime}(4)=6$$ to approximate $$f(3.8)$$.

Problem 5

Assume that the function $$f$$ is defined for $$x\geq 0$$ in the following way: $$\displaystyle f(x)=\frac{4x-2}{x+3}$$. Determine the inverse function of $$f$$.

Problem 6

If $$\displaystyle x^{6}e^y+5=xy^{3}$$, determine $$\displaystyle\frac{dy}{dx}$$.

Problem 7

Find the derivative of $$\displaystyle y=\ln\left(\frac{(x-8)^{\frac1{2}}(x+3)}{x}\right)$$.

Problem 8

Find the derivative of the function $$\displaystyle f(x)=x^{\ln\left(x^{8}\right)}$$.

Problem 9

The value of the expression $$x\cdot y$$, with both variables positive, is $$12$$. What is the smallest value of $$3x+4y$$?

Problem 10

Sketch the graph of a function $$f$$ that has the following characteristics: \begin{eqnarray*} &&f(2)=f(7)=0\newline && f^{\prime}(x)> 0 \;\mbox{for }x< 5\newline &&f^{\prime}(5) \;\mbox{does not exist}\newline && f^{\prime}(x)< 0 \;\mbox{for }x> 5\newline && f^{\prime\prime}(x)> 0\;\mbox{for }x\neq 5. \end{eqnarray*}

Problem 11

The triangle $$ABC$$ has right angle at vertex $$A$$. The sides of the triangle satisfy $$AB=12$$ and $$AC=6$$. What is the maximal possible area of a rectangle that can be inscribed in the triangle $$ABC$$ if one of its vertices is $$A$$ and another vertex $$P$$ must belong to the edge $$BC$$?

Problem 12

Evaluate the integral $$\displaystyle\int_{4}^{6}\frac{2\sqrt{x-1} +4}{x\sqrt{x-1} }\,dx$$.

Problem 13

Consider the functions $$f(x)=x^3$$ and $$g(x)=16x-6x^2$$.

• (a) Sketch the region $$R$$ in the $$xy$$ plane that is located between the graphs of the functions $$f$$ and $$g$$ for $$x\geq 0$$.

• (b) Evaluate the area of the region $$R$$ from the previous part of the problem.

Problem 14

Do there exist two functions $$f:\mathbb R\to\mathbb R$$ and $$g:\mathbb R\to\mathbb R$$ that are both onto but the function $$h:\mathbb R\to\mathbb R$$ defined as $$h(x)=f(x)+12g(x)$$ is not onto? Explain your answer.

The symbol $$\mathbb R$$ denotes the set of all real numbers.

Problem 15

Prove that for every differentiable function $$f$$ such that $$\displaystyle\int_0^{ 7 }f(t)\,dt=\int_0^{7}e^{-t^4}\,dt$$ there exists $$c$$ that satisfies $$0\leq c\leq 7$$ and $$e^{c^4}\cdot f(c)=1$$.