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Section Practice

These are practice problems for the tests. Solutions are in the next section. While we will not present these, I am happy to answer questions about them in class.

Graphing

  1. Graph \(f(x) = x^2 - 4\text{,}\) listing \(x-\) and \(y-\)intercepts.

  2. Graph \(\dsp g(x) = \frac{x-1}{x+2}\text{,}\) listing both vertical and horizontal asymptotes.

  3. Graph \(h(x) = \sin(x)\text{,}\) listing the domain and range.

  4. Graph \(r(x) = \sqrt{3x-2}\text{,}\) listing the domain and \(x-\)intercept.

  5. Graph \(\dsp f(x) = \left\{ \begin{array}{ll} x-5 \amp \mbox{if} \; \; \; \; x>1 \\ x^{2}+1 \amp \mbox{if} \; \; \; \; x\leq 1 \end{array} \right.\)

Evaluate the following limits or state why they do not exist.

  1. \(\lim_{x\to 2} (x^{3}+2x^{2}-5x+7)\)

  2. \(\dsp \lim_{x\to \frac{\pi}{2}}\cos(2x)\)

  3. \(\dsp \lim_{x\to -2}\frac{x+2}{x^{2}-4} \;\;\; \mbox{and} \;\;\; \lim_{x\to -2}\frac{x^{2}-4}{x+2}\)

  4. \(\lim_{x\to 1} f(x)\text{,}\) where \(\dsp f(x) = \left\{ \begin{array}{ll} x-5 \amp \mbox{if} \; \; \; \; x>1 \\ x^{2}+1 \amp \mbox{if} \; \; \; \; x\leq 1 \end{array} \right.\)

  5. \(\lim_{x\to 3}\sqrt{x-3}\)

  6. \(\dsp \lim_{h\to 0}\frac{3xh+h}{h}\)

  7. \(\dsp \lim_{x\to -3} \frac{x^{2}+5x+6}{x^{2}-x-12}\)

  8. \(\dsp \lim_{t\to 1} \frac{t^{3}-1}{t^{2}-1}\)

  9. \(\dsp \lim_{x\to -1} \frac{1+2x}{1+x}\)

  10. \(\dsp \lim_{x\to 0} \frac{\sqrt{3-x}-\sqrt{3}}{x}\)

  11. \(\lim_{x\to \frac{\pi}{8}^{+}}\cos(2x)\text{,}\) \(\lim_{x\to \frac{\pi}{8}^{-}}\cos(2x)\text{,}\) and \(\lim_{x\to \frac{\pi}{8}}\cos(2x)\)

  12. \(lim_{x\to 1^{-}}f(x)\text{,}\) \(lim_{x\to 1^{+}}f(x)\text{,}\) and \(lim_{x\to 1}f(x)\) where \(\dsp f(x) = \left\{ \begin{array}{ll} x+3 \amp \mbox{if} \; \; \; \; x>1 \\ 9x^{2}-5 \amp \mbox{if} \; \; \; \; x\leq 1 \end{array} \right.\)

  13. Let \(f(x)=1-2x^2\text{.}\) Compute \(\dsp \lim_{t \to x} \frac{f(t)-f(x)}{t-x}\text{.}\)

  14. Use the Squeeze Theorem to compute \(\dsp \lim_{x \to \infty} \frac{\sin(x) + 2}{x}\text{.}\)

List the interval(s) on which each of the following is continuous.

  1. \(\dsp f(x)=\frac{3x+2}{6x^{2}-x-1}\)

  2. \(G(x)=\sqrt{x+5}\)

  3. \(f(x)=x^{4}+2x^{3}-6x+1\)

  4. \(h(x)=\sqrt{2-4x}\)

  5. \(\dsp g(t)=\frac{t}{t-2}\)

  6. \(f(\beta)=3\tan (\beta)\)

  7. \(\dsp f(x) = \left\{ \begin{array}{ll} x^{2}+2 \amp \mbox{if} \; \; \; \; x \geq 1 \\ 5x-2 \amp \mbox{if} \; \; \; \; x \lt 1 \end{array} \right.\)

  8. \(f(x) = \left\{ \begin{array}{ll} 3-x^{2} \amp \mbox{if} \;\;\;\;x\geq \pi \\ \cos(2x) \amp \mbox{if} \;\;\;\; x\lt \pi \end{array} \right.\)

Velocity

  1. Given position function \(f(t)=3t^{2}+t-1\text{,}\) where \(t\) is in seconds and \(f(t)\) is in feet, find the average velocity between \(1\) and \(4\) seconds. Find the average velocity between \(p\) and \(q\) seconds.

  2. Suppose that \(y(t)=-5t^{2}+100t+5\) gives the height \(t\) (in hundreds of feet) of a rocket fired straight up as a function of time \(t\) (in seconds). Determine the instantaneous velocity at times \(t=2\) and at \(t=4\text{.}\)

Secant and Tangent Lines

  1. Find the equation of the secant line of \(g(x)=1+\sin(x)\) containing the points \((0,1)\) and \((\frac{\pi }{2},2)\text{.}\)

  2. Find the slope of the tangent line to \(f(t)=5t-2t^{2}\) at \(t=-1\text{.}\)

  3. Find the equation of the tangent line to \(\dsp F(x) = \frac{x}{x+1}\) at \(x=-3.\) Graph both \(F\) and the tangent line.

  4. Find the equation of the tangent line to \(H(x)=\sqrt{1+x}\) at \(x=8\) and sketch the graph of both \(H\) and the tangent line.

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