MATH 16A

Midterm 1 | Winter '16 | Gravner

(a) A line has $x$-intercept $(4,0)$ and slope $-1/2$. Find its $y$-intercept.

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(b) A circle has center $(0,2)$ and goes through the point $(1,4)$. Find the equation of this circle.

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(c) Find all points of intersection (if there are any) between the line from (a) and the circle from (b).

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Consider the function $f(x) = \frac{x^2 - 4x}{x^2 - 4}$. Determine the domain, intercepts, and vertical and horizontal asymptoes. (Include computation of limits at vertical asymptoes.) Determine also any points where the graph of $y = f(x)$ intersects its horizontal asymptote. Then sketch the graph of this function on which all obtained points and asymptotes are clearly marked.

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Compute the following limits. Give each answer as a finite number, +$\infty$, or -$\infty$
(a) $\lim_{x \to 4} \frac{\sqrt{x + 5}}{\sqrt{x - 3}}$

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(b)

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(c)

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In all parts of this problem,
(a) Determine the domain of the function $y = f(x)$.

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(b) Compute $g(f(1))$.

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(c) A line is tangent to the graph of $y = f(x)$ and perpendicular to the line $x+3y+7=0$. Determine the equation of this line (in the slope-intercept form).

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(d) Discuss continuity and differentiability of $y = g(x)$, and determine the range of this function.

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