Math 1A: Discussion ExercisesGSI: Theo Johnson-Freydhttp://math.berkeley.edu/∼theojf/09Spring1A/Find two or three classmates and a few feet of chalkboard. Introduce yourself to your newfriends, and write all of your names at the top of the chalkboard. As a group, try your hand atthe following exercises. Be sure to discuss how to solve the exercises — how you get the solutionis much more important than whether you get the solution. If as a group you agree that you allunderstand a certain type of exercise, move on to later problems. You are not expected to solve allthe exercises: in particular, the last few exercises may be very hard.Many of the exercises are from Single Variable Calculus: Early Transcendentals for UC Berkeleyby James Stewart; these are marked with an §. Others are my own, or are independently marked.More Preview and Review1. § Classify each function as a power function, root function, polynomial (state its degree),rational function, algebraic function, trigonometric function, exponential functions, or loga-rithmic function. Some functions are more than one of these, and some are none.(a)5√x(b) x9+ x4(c) tan 2x(d) (sin x)2(e)p1 − x2(f)x2+ 1x3+ x(g) log10x(h) ln sin xRemark: We will develop powerful techniques with which to manipulate (in particular,differentiate and integrate) all these functions. Exactly what techniques we will use willdepend on the type of function. For example, the derivative (we haven’t defined this yet) ofa power function isD[xa] = a × xa−1(1)whereas the derivative of an exponential function isD[ax] = ln a × ax. (2)One of the most common mistakes in Math 1A is to use the formula from equation (1) when(2) is correct. The best way to avoid the mistake is to get very good at telling what type offunction you’re dealing with, and understanding how the different types are different.2. (a) § What do all members of the family of linear functions f (x) = 1 + m(x + 3) have incommon? Sketch several members of the family.(b) § What do all members of the family of linear functions f (x) = c − x have in common?Sketch several members of the family.(c) Which functions (if any) belong to both families?3. § If the recommended adult dosage for a drug is D (in mg), then to determine the appropriatedosage c for a child of age a, pharmacists use the equation c(a) = 0.0417D(a + 1).Remark: c is a function of both a and D, but for the purposes of this problem we’re thinkingof a as the independent variable and c as the dependent variable, so that c is a function of a,and we’re thinking of D as a “fixed unknown constant”, which is actually just another wordfor “variable”.1(a) What does the slope of the graph of c(a) represent? If the dosage for an adult for thisparticular drug is 200 mg, what is this slope?(b) What is the dosage for a newborn in terms of D? What feature of the graph of c(a)corresponds to this amount?(c) What is the domain of the function c(a)? This is a rather tricky question: think aboutthe real-word set-up. For instance, is a = 100 in the domain? What about a = π?4. Kepler’s Third Law of Planetary Motion states thatThe square of the period of revolution of a planet is proportional to the cube of itsmean distance from the sunLet’s use units so that we measure mean distance in Astronomical Units (AU), defined sothat the mean distance from the Earth to the Sun is 1 AU (1 AU is 149 598 000 kilometers),and so that we measure periods of revolution in years, defined so that the period of revolutionof the Earth is 1 year.(a) Choose variables to represent all quantities in the above Third Law. Rewrite the ThirdLaw as an equation.(b) Solve this equation to express the mean distance as a function of the period of revolution,and also solve it to express the period of revolution as a function of the mean distance.What kinds of functions are these?(c) Mars is roughly 1.5 AU from the Sun. What is its period of revolution?Saturn’s period of revolution is just about 30 years. What is its mean distance?5. § Suppose you are given the graph of y = f(x). Write equations (in terms of f) for thefunctions whose graphs are obtained from f as follows:(a) Shifted upward by 3.(b) Shifted leftward by 3.(c) Shifted rightward by 3.(d) Reflected about the x-axis.(e) Reflected about the y-axis.(f) Reflected about the origin.(g) Stretched vertically by a factor of 3.(h) Stretched horizontally by a factor of 3.(i) Stretched in both directions by a factor of 3.(j) Shrunk vertically by a factor of 3.(k) Shrunk horizontally by a factor of 3.(l) Shrunk in both directions by a factor of 3.6. § To the right is the graph of y = f (x). Use itto graph the following functions:(a) y = f (2x)(b) y = f (x + 2)(c) y = f (−x)(d) y = −f (−x) + 1117. § How is the graph of y = f (|x|) related to the graph of y = f (x)? Sketch the graphs ofy = sin |x|, y =p|x|, and (most importantly for our class) y = ln |x|.8. § Suppose that f and g are even functions. What can you say about f + g and f g? What iff and g are both odd?9. Use algebra to show the shifting a graph by a units upward and then stretching verticallyby a factor of b is the same as first stretching the graph vertically by a factor of b and thenshifting upward by