#### MATH 2A

##### Final - Practice 1 | Fall '16| Campbell
1. For what value of $a$ is the function $f(x) = \begin{cases} x^2 & x < 3 \\ 2ax & x \ge 3 \end{cases}$ continuous at every $x$?

2. The theory of relativity predicts that an object whose mass is $m_0$ when it is at rest will appear heavier when it is moving at speeds near the speed of light. When the object is moving at speed $v$, its mass $m$ is given by $m = \frac{m_0}{\sqrt{1 - (v^2/c^2)}}$ where $c$ is the speed of light. Find $\frac{dm}{dv}$ and explain in terms of physics what this quantity tells you.

3. Show that the equation $3x + 2 \cos x + 5 = 0$ has exactly one real root.

4. A hyperbola is given by th equation $x^2 + 2xy - y^2 + x = 2$. Use implicit differentiation to find an equation of the tangent line to this curve at the point $(1, 2)$.

5. Little Susie is enjoying a nice spherical lollipop. She sucks the lollipop in such a way that the circumference decreases by $1$ centimeter per minute. How fast is the volume of her lollipop changing when the lollipop has a radius of $5$ centimeters?
6. Find the linear approximation of the function $f(x) = x^{3/4}$ at the point $a = 16$.
7. If $f(3) = 4, g(3) = 2, f'(3) = -5, g'(3) = 6$, find the following values
a) $(f + g)'(3) Log in or sign up to see discussion or post a question. b)$(fg)'(3)$Log in or sign up to see discussion or post a question. c)$\left( \frac{f}{g} \right)'(3)$Log in or sign up to see discussion or post a question. 8. Find the absolute maximum and minimum values of the function$f(x) = 3x^4 - 4x^3$on the interval$[-1, 2]$. Log in or sign up to see discussion or post a question. 9. A balloon ascending at a rate of$12$ft/s is at a height of$80$ft above the ground when a package is dropped. How long does it take the package to reach the ground? (Hint: the acceleration due to gravity is$32 \; ft/s^2$downward. Use antiderivatives.) You may leave your answer in radical form. Log in or sign up to see discussion or post a question. 10. The graph of$f(x)$is below. Sketch graphs for$f'(x)$and$f''(x)$. Log in or sign up to see discussion or post a question. 11. Complete each of the following definitions and statements. a) A function$f$is continuous at a number$a$if ________. Log in or sign up to see discussion or post a question. b) The derivative of a function$f$at a number$a$is$f'(a) =$________ if this limit exists. Log in or sign up to see discussion or post a question. c) The Intermediate Value Theorem says ________ Log in or sign up to see discussion or post a question. d) ________ says "If$f$is continuous on a closed interval$[a, b]$, then$f$attains an absolute maximum value$f(c)$and an absolute minimum value$f(d)$at some numbers$c$and$d$in$[a, b]$." Log in or sign up to see discussion or post a question. e) A function$F$is called an antiderivative of$f$on an interval$I$if ________ for all$x$in$I$. Log in or sign up to see discussion or post a question. 12. Find the dimensions of the rectangle with largest area that can be inscribed in a semicircle of radius$2$inches. Log in or sign up to see discussion or post a question. 13. For the following problems, find the limit if it exists or explain why the limit does not exist. a)$\lim_{x \to 0} \frac{\sqrt{x^2 + 9} - 3}{x^2}$Log in or sign up to see discussion or post a question. b)$\lim_{x \to \infty} \frac{5x + 2}{7x^2 - 4x + 8}$Log in or sign up to see discussion or post a question. c)$\lim_{x \to 3^-} \frac{x}{x - 3}$Log in or sign up to see discussion or post a question. d)$\lim_{x \to 1} \frac{x - 1}{x^4 - 1}$Log in or sign up to see discussion or post a question. e)$\lim_{x \to 1} \frac{x^2 - 1}{|x - 1|}$Log in or sign up to see discussion or post a question. 14. Compute each of the following. a)$\frac{dy}{dt}$for$y = \frac{1}{\sqrt{t}} + 5t + 3e^t$Log in or sign up to see discussion or post a question. b)$f'(4)$for$f(x) = \sqrt{9 + 4x}$Log in or sign up to see discussion or post a question. c)$f'(x)$for$f(x) = \sin(x \tan^{-1} x)$Log in or sign up to see discussion or post a question. d)$h'(r)$for$h(r) = r \ln 3r$Log in or sign up to see discussion or post a question. e)$\frac{dy}{dx}$for$y = x^{\tan x}$. Log in or sign up to see discussion or post a question. 15. Consider the function$f(x) = \frac{(x + 1)^2}{1 + x^2}$a) Find the domain of$f(x)$. Log in or sign up to see discussion or post a question. b) Find the$x$and$y$intercepts. Log in or sign up to see discussion or post a question. c) Determine if$f(x)$is even, odd, or periodic. Log in or sign up to see discussion or post a question. d) Find any vertical, horizontal or slant asymptotes of$f(x)$. Log in or sign up to see discussion or post a question. e) Find intervals on which$f(x)$is increasing and on which it is decreasing. Log in or sign up to see discussion or post a question. f) Find any local maximum and minimum values. Log in or sign up to see discussion or post a question. g) Find intervals on which$f(x)$is concave up and on which it is concave down. Log in or sign up to see discussion or post a question. h) Find any points of inflection. Log in or sign up to see discussion or post a question. i) Sketch a graph of$f(x)\$.