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# MTH 2610: Midterm 1 Practice 1

Problem 1

Consider the functions $$f(x)=\sqrt x$$ and $$g(x)=11x-18$$. Determine $$f(g(9))$$.

Problem 2

Evaluate the limit $$\displaystyle \lim_{x\to 0}\frac{12x^3+5x^2+15x^4}{x^4+x^3+x^2}$$.

Problem 3

Determine the following limits:

• (a) $$\displaystyle \lim_{x\to 7} \frac{x^2-4x-21}{x-7}$$;

• (b) $$\displaystyle \lim_{x\to 0}\frac{x^2}{\sqrt{11+9x^2}-\sqrt{11}}$$.

Problem 4

Evaluate the limit $$\displaystyle \lim_{x\to 0}\frac{\frac1{1+4x}-\frac1{1+x^2}}{ 7x}$$.

Problem 5

Consider the function $$\displaystyle f(x)=8x^2-5x-1$$.

• (a) Determine $$f^{\prime}(1)$$.

• (b) Determine the equation of the tangent line to the graph of $$f$$ at the point $$(1,2)$$.

Problem 6

Evaluate the derivatives of the following functions:

• (a) $$\displaystyle f(x)=\left(x+7\sin x\right)\left(\cos x-3\right)$$;

• (b) $$\displaystyle f(x)=\frac{x-7}{x^2+3}$$;

• (c) $$\displaystyle f(x)=\frac{ \left(x+7\right)\left(x-3\right)}{x^2+\sin x}$$.

Problem 7

Assume that $$f$$ is a function such that $$f^{\prime}(x)=0$$ for all $$x$$. Assume that $$g$$ is any function. Which of the following functions could have a slope that is different from 0?

• (A) $$\displaystyle 7f(x)$$

• (B) $$\displaystyle f\left(x^2\right)\cdot f(g(x))+f(f(x))$$

• (C) $$\displaystyle x^3g(x^2)-x^3g(f(x))$$

• (D) $$\displaystyle f(g(x))$$

• (E) $$\displaystyle x^2f(x)-x^2f(g(x))$$

Circle the letter in front of the correct function, and provide mathematically rigorous justification for your answer.