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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.