Math 1A: Discussion ExercisesGSI: Theo Johnson-Freydhttp://math.berkeley.edu/∼theojf/09Spring1A/Find two or three classmates and a few feet of chalkboard. As a group, try your hand at thefollowing exercises. Be sure to discuss how to solve the exercises — how you get the solution ismuch 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.Derivatives1. § For each of the following functions, find f0(a). You may use only: the definition of thederivative, and Derivative Laws allowed on next week’s midterm (sum, difference, constantmultiple, power, exponential).(a) f(t) =2t + 1t + 3(b) f(x) =1√x + 2(c)√3x + 1 (d) f(x) = 3−2x+4x22. § Make a careful sketch of the graph of y = sin x, and sketch the graph of the derivative sin0x.In particular, what are the zeros of sin0, and where is it positive and negative. Can you guessthe formula for sin0based on the graph?3. What is the domain of the function f(x) =√x? What is the domain of its derivative f0(x)?4. § Suppose that f is a function with the property that |f(x)| ≤ x2for all x. Show thatf(0) = 0. Then show that f0(0) = 0.5. § Differentiate the following functions, using only the Derivative Laws allowed on the midterm.(a) f(x) =√30(b) f(t) =12t6− 3t4+ t(c) g(t) =14t4+ 8(d) f(x) = 5ex+ 3(e) y =3√x(f) h(x) =x2− 2√xx(g) F (x) =12x5(h) y = ax2+ bx + c(i) v =3√x +1√x26. § Find equations for the tangent line and the normal line to the curve y = (1 + 2x)2at thepoint (1, 9).7. § Graph the function f(x) = x +1x, and also find and graph the derivative F0(x). Are yourgraphs consistent?8. § Find all derivatives of the function f(x) = x4−3x3+ 16x. I.e. find the first derivative, thesecond derivative, etc., until you get some derivative that is identically 0.9. What happens if you take the function 1/x and start differentiating? Does the sequence offunctions ever stop, in the sense of eventually becoming identically zero? Justify your answer.10. § For what values of x does the graph of y = x3+ 3x2+ x + 3 have a horizontal tangent?111. § Show that the curve y = 6x3+ 5x − 3 has no tangent line with slope 4.12. § Find an equation of the tangent line to the curve y = x√x that is parallel to the liney = 1 + 3x.13. § Find equations of both lines that are tangent to the curve y = 1 + x3and are perpendicularto the line x + 12y = 1.14. § Draw a picture to show that there are two tangent lines to the parabola y = x2that passthrough the point (0, −4). Find the coordinates of the points where these tangent lines meetthe parabola.15. (a) § Find equations of both lines through the point (2, −3) that are tangent to the parabolay = x2+ x.(b) § Provide two proofs, one using algebra and the other using a graph, to show that thereis no line through the point (2, 7) that is tangent to the parabola.16. § Find a second-degree (i.e. quadratic) polynomial P such that P (2) = 5, P0(2) = 3, andP00(2) = 2.17. § Letf(x) =x2if x ≤ 2mx + b if x > 2For what values of m and b is f(x) differentiable everywhere?18. Find the derivative of the function f(x) = x · |x|. Be sure to specify at what points f isdifferentiable.Product and Quotient Rules19. § Suppose that f (5) = 1, f0(5) = 6, g(5) = −3, and g0(5) = 2. Find (fg)0(5), (f/g)0(5), and(g/f)0(5).20. § Differentiate. You may use the product and quotient rules.(a) (x3+ 2x)ex(b) (u−2+ u−3)(u5− 2u2)(c)x + 1x3+ x − 2(d)t(t − 1)2(e)2t4 + t2(f)ax + bcx + d21. § Find f0(x) and f00(x):(a) f(x) = x4ex(b) f(x) = x5/2ex(c) f(x) =x21 + 2x22. § How many tangent lines to the curve y = x/(x + 1) pass through the point (1, 2)? At whatpoints do these tangent lines touch the curve?23. § Use the Product Rule twice to prove that if f, g, h are differentiable, then (fgh)0= f0gh +fg0h + fgh0. Then take f = g = h to show thatddxf(x) 3= 3f(x) 2f0(x), and use this todifferentiate y = e3x.24. § If f and g are differentiable, show that (fg)00= f00g + 2f0g0+ fg00. Find similar formulasfor (fg)000and (fg)(4). Do you notice a pattern? Guess a formula for