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.Introducing Derivatives1. § If a ball is through into the air with a velocity of 40 ft/s, its heigh in feet t seconds later isgiven by y = 40ft/st −16ft/s2t2.(a) Find the average velocity for the time period beginning when t = 2 and lastingi. 0.5 seconds ii. 0.1 seconds iii. 0.05 seconds iv. 0.01 seconds(b) Estimate the instantaneous velocity when t = 2.2. § The point P = (1,12) lies on the curve {y = x/(1 + x)}.(a) Let Q = (x, x/(1 + x)). In terms of x, find the slope of the secant line P Q. What is thedomain of your function? Why?(b) Using your answer to part (a), find the value of the slop of P Q for the following valuesof x.i. 0.5ii. 1.5iii. 0.9iv. 1.1v. 0.99vi. 1.01vii. 0.999viii. 1.001You should report numerical answers rounded to a few decimal places. One way to dothis is to use a calculator. Another way is find the reciprocal of the slope, and then usethe fact that, if |b|  |a| (b is much smaller than a), then1a + b≈1a−ba2(1)(c) If you plug x = 1 into your formula from part (a), what value of the slope would youget? Does this number agree with your answers to part (b)?(d) Find an equation of the tangent line to the curve at P = (1,12).3. Let c be a real constant and f(x) a function. How to the slopes of secant and tangent linesof g(x) = f(x) + c compare to those of f (x)? What about h(x) = f(x + c)?4. For each of the following functions, calculate the difference quotientf(x + h) − f(x)h. Simplifyyour answers.(a) f(x) = x(b) f(x) = x2(c) f(x) = x3(d) f(x) = xn(e) f(x) = 1/x(f) f (x) = 1/x2(g) f(x) = 1/xn(h) f(x) =√x1(We always let n be a fixed unknown positive integer.) What do difference quotients have todo with secant and tangent lines?5. (a) Let |h|  |x|, so that h is much smaller than x. Simplify the difference quotient for thenatural logarithm:ln(x + h) − ln(x)hYou should use rules about logarithms, and you should also use the approximation, truewhen c is very close to 0, thatln(1 + c) ≈ c. (2)(b) When |h|  |x|, simplify the difference quotient for the function ex. Exponentiateequation (2) for a useful approximation.ec≈ 1 + c.6. Let f and g be two functions, and a and b two numbers. Let l be the slope of the secant tof over the domain [a, b] — i.e. l is the slope of the line that goes through the points (a, f(a))and (b, f(b)) — and let m be the slope of the secant to g over the domain [a, b]. Let p be theslope of the secant to the function h = f + g over [a, b]. Prove that p = l + m.7. Prove equation (1):(a) In terms of a and b, evaluate the “error” of the approximation:1a + b−1a−ba2(b) Argue that when b is much smaller than a, this error is very small.8. In Math 1B we will develop tools to study approximations like (1). For example, we cangeneralize (1) to:1a + b≈1a−ba2+b2a3− ··· ±bnan+1(3)where the last term is “+” if n is even, and “−” if n is odd. When |b| < |a|, for large enoughn the fraction bn/an+1is very small.Let’s prove equation (3), in the case when |b| < |a|:(a) Begin by multiplying both sides of the equation by (a + b), and simplify the right-hand-side. What is the error of the new approximation?(b) You may use the following fact: if |c| < 1, and d is any positive number, then for somen, |cn| < d. Show that for any d, there is some n such that the error from part (a) isless than d.(c) If n gets larger, what happens to the error?Hard problems from previous days9. § 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|.10. 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