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AP Calculus AB

  1. 1.

    Suppose the function $$f$$ has the graph as shown above. Which of the following limits exist?

    1. $$\lim\limits_{x \to a}f(x)$$
    2. $$\lim\limits_{x \to b}f(x)$$
    3. $$\lim\limits_{x \to c}f(x)$$
    1. Limit exists because the limit is independent of the actual output $$f(a)$$. It does not matter where the point actually is at $$x=a$$
    2. Limit does not exist because $$\lim\limits_{x\to b^-}f(x)\not=\lim\limits_{x\to b^+}f(x)$$
    3. Limit exists because it is a simple curve
  2. 2.

    Let $$f$$ be the function defined by $$f(x)=x^3+x-4$$. What is the value of $$c$$ for which the instantaneous rate of change of $$f$$ at $$x=c$$ is the same as the average rate of change of $$f$$ over $$[-1,3]$$?

    $$$ \begin{aligned} f^\prime(x)&=3x^2+1\\ \text{Ave slope }&=\frac{f(3)-f(-1)}{3-(-1)} = \frac{26-(-6)}{4}=\frac{32}{4}=8\\\\ \text{We need } f^\prime(x)&=3x^2+1=8\\ x&=\sqrt{\frac{7}{3}} \end{aligned} $$$
  3. 3.

    $$$ \int \frac{x}{(3x^2+5)^4}\,dx=$$$

    $$$ \begin{aligned} \int \frac{x}{(3x^2+5)^4}\,dx\\ \text{Use substitution: } u&=3x^2+5\\ du&=6x\,dx\\ \frac{1}{6}\,du&=x\,dx\\ \text{So }\int \frac{x}{(3x^2+5)^4}\,dx&= \frac{1}{6}\int \frac{1}{u^4}\,du\\ &=\frac{1}{6}\int u^{-4}\,du\\ &=\frac{1}{6}\frac{u^{-3}}{-3}+C\\ &=\frac{-1}{18(3x^2+5)^3+C} \end{aligned} $$$