MA123, Chapter 4: Computing Some Derivatives (pp. 69-82, Gootman)Chapter Goals:• Understand the derivative as the slop e of the tangent line at a point.• Investigate further the notions of continuity and differentiability.• Use the definition to calculate some derivatives.• Use the definition to approximate some derivatives.Assignments:Assignment 06 Assignment 07In this chapter we explore further the relation between the derivative and the equation of the tangent line ata point. Then we learn how to compute the derivative of some functions using the defi nition of the derivative.One reason for doing this is to convince you that the rules and formulas for derivatives are not magical. T heyhave a solid foundation and can be explained with just a little bit of effort. Learning should not ju st be amatter of memorizing mysterious formulas but it should rather be a matter of understanding them.We start by recalling the following facts that we encountered in Chapter 2:◮ Basic facts about derivatives:The instantaneous rate of change of a function f with respect to x ata general point x is called the derivative of f at x and is denoted with f′(x):f′(x) = limh→0f(x + h) − f (x)h.For a given value x0, the derivative of f at x0, namely f′(x0), gives the slope of the tangent line to the graphof f at the point (x0, f(x0)). Thus, the equation of the tangent line to such a point is given by the formulay = f(x0) + f′(x0)(x − x0).◮ Tangent lines, continuity and differentiability: In the following pr ob lems we practice computingequations of tangent lines. Also, we investigate further the notions of continuity and differentiability of afunction at a point. Please refer back to Chapter 3 f or the corresponding definitions.Example 1: The graph of a function h(x) and the coordinates of a point (x0, h(x0)) on the graphs of h aregiven below. Find h′(x0) by analyzing the graph.xy34h′(3)?=xy33h′(3)?=Note: In the following problems you can use the fact that the derivative of f(x) = ax2+ bx + c isf′(x) = 2ax + b. (See the calculation carr ied out in Chapter 2, Example 15.)31Example 2: Consider the function f(x) = 3x2−6x −10. Write the equation of the tangent line to thegraph of f at x = −2 in the form y = mx + b, for ap propriate constants m and b.Example 3:Consider the function g(x) = −3x2+ 7x −6. Write an equation of the tangent line to thegraph of g at x = 1. For which values of y1and y2does this tangent line go through the points (−1, y1)and (4, y2)?Example 4:Suppose that the equation of the tangent line to the graph of the function f(x) =√x + aat x = 16 is given by y = mx + 5. Find a and m.Hint: You may use f′(x) =12√x.32Example 5: Determine th e x values where the derivative of the function is n ot defined (that is the pointswhere the function is not differentiable). Is the function continuous at those points?g(x) =−x + 1 if x ≤ −3|x| if − 3 < x < 2−x2+ 6 if x ≥ 2xy−30 2234Example 6:Determine the x values where the derivative of h(x) = |x2− 7x + 10| is not defined. Is h(x)continuous at those points? (Hint: first draw the graph of the equation y = x2− 7x + 10 and then draw thegraph of the function h.)Next, we use the definition of the derivative to learn how to differentiate functions of the following types:f(x) = (x + α)2f(x) =1x + αf(x) =√x + α f(x) = (x + α)3where α is an arbitrary real number . For each type of function, the calculation of the limit has to be treatedwith a different technique.Example 7:Let f (x) = (x + 4)2.(a) Find constants A, B, and C such thatf(x + h) − f(x)h= Ax + Bh + C.(b) Show that the derivative of f is given by the expression f′(x) = 2x + 8 = 2(x + 4).(c) Find f′(5). Write th e equation of the tangent line to the graph of f at x = 5 in th e form y = mx + b.33Note: Expanding the binomial (x + 4)2in the expression for the function f of Example 6 yields thatf(x) = x2+ 8x + 16. Hence the result f′(x) = 2x + 8 also follows from the calculation carried out in Ch ap ter2, Example 15.Example 8:Let f(x) =1x + 3.(a) Find constants A, B, C, and D such thatf(x + h) − f (x)h=A(x + B)(x + Ch + D).(b) Show that the derivative of f is given by the expression f′(x) =−1(x + 3)2= −(x + 3)−2.(c) Find f′(5).Example 9:Let f(x) =√x − 2.(a) Find constants A, B, C, and D such thatf(x + h) − f (x)h=A√Bx + Ch + D +√x − 2.(b) Show that the derivative of f is given by the expression f′(x) =12√x − 2=12(x − 2)−1/2.(c) Find f′(6) and f′(11).34Special product formulas: The powers of certain binomials occur so frequently that we shouldmemorize the following formulas. We can verify them by performing the multiplications.If A and B are any real numbers or algebraic expressions, then:(1.) (A + B)2= A2+ 2AB + B2(2.) (A − B)2= A2− 2AB + B2(3.) (A + B)3= A3+ 3A2B + 3AB2+ B3(4.) (A − B)3= A3−3A2B + 3AB2− B3Visualizing a formula:Many of the special product formulas can be seen asgeometrical facts about length, area, and volume. Theancient Greeks always interpreted algebraic formulas interms of geometric figures.For example, the figure belowAB(A + B)2A + BA + BA B••••ABA2ABABB2A B••••(A + B)2= A2+ 2AB + B2shows how the formula for the square of a binomial(formula 1) can be interpreted as a fact about areas ofsquares and rectangles.Pascal’s triangle:The coefficients (without sign)of the expansion of a binomial of the form (a ± b)ncan be read off the n-th row of the following ‘triangle’named Pascal’s triangle (after Blaise Pascal, a 17thcentury French mathematician and philosopher).To build the triangle, start with ‘1’ at the top, thencontinue placing numbers below it in a triangular way.Each number is simply obtained by adding the twonumbers d irectly above it.11 11 2 11 3 3 11 4 6 4 11 5 10 10 5 1n = 0n = 1n = 2n = 3n = 4n = 5...Example 10:Let g(x) = (x − 4)3.(a) Find constants A, B, C, D, E and F such thatg(x + h) −g(x)h= Ax2+ Bx + C + Dxh + Eh + F h2.(b) Show that the derivative of g is given by the exp ression g′(x) = 3(x − 4)2= 3x2− 24x + 48.(c) Find g′(6) and g′(−1).35Example 11: If f(x) =−2x − 3, thenf(x + h) − f (x)h=A(x − 3)(x + Bh + C). Find A, B, and C.Example 12:Suppose thatf(x + h) − f(x)h=−2h(x + 2) − h2h(x + h + 2)2(x + 2)2.Find the slope m of the tangent line at x = 1.Example 13:Suppose thatf(x + h) − f(x)h= 3x + 2h − 1 and f(1) = 4.Find the equation of the tangent line to the graph