Bob Brown Math 251 Calculus 1 Chapter 3, Section 1 Completed 1 CCBC Dundalk

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1 Bob Brown Mat 251 Calculus 1 Capter 3, Section 1 Completed 1 Te Tangent Line Problem Te idea of a tangent line first arises in geometry in te context of a circle. But before we jump into a discussion of tangent lines, we begin by considering secant lines. A secant line passes troug a circle in two places. Te following are examples of a secant line troug a circle. Secant Lines troug Circles A tangent line touces a circle in just one place. Te following are examples of a tangent line to a circle. Tangent Lines to Circles Wat Do Secant Lines and Tangent Lines Have To Do Wit Calculus? We can calculate te slope of te line segment connecting te two points at wic te secant line intersects te circle. Now, old te left point (for example) fixed and move te rigt point toward te left point, redrawing a secant line eac time and calculating te slope of tat secant line. In te it, we ave a tangent line at a point wose slope is te it of te slopes of tose secant lines. In Calculus, we often work wit te grap of a function y = f(x). We can calculate te slope of te line segment connecting two points on te grap of y = f(x). y = f(x) Now, old te left point (for example) fixed and move te rigt point toward te left point, redrawing a secant line eac time and calculating te slope of tat secant line. y = f(x) y = f(x) y = f(x) In te it, we ave a tangent line at a point wose slope is te it of te slopes of tose secant lines.

2 Bob Brown Mat 251 Calculus 1 Capter 3, Section 1 Completed 2 Determining a Derivative Function Grapically Exercise 1: Sketc te tangent lines to y = f(x) at x-values corresponding to te integers from -2 to 5, inclusive, and estimate teir slopes, recording your estimates in te table below. Your estimates may be sligtly different tan mine. Note tat te x-axis is scaled by ones and te y-axis is scaled by fives. x f (x) f (x) is te notation for te slope of te tangent line to te grap of f at x. For example, f (3) is te notation for te slope of te tangent line to te grap of f at x = 3. By filling in te table, you are creating a function, te derivative function of te original function f sketced above. Plot tis table below, generating a grap of y f (x).

3 Bob Brown Mat 251 Calculus 1 Capter 3, Section 1 Completed 3 Determining a Derivative Function Algebraically Def.: For any function y = f(x), we define its derivative function y f (x) by f (x) = 0 f ( x ) Note tat te text uses x instead of. Question: Were does tis formula come from? Answer: Te fraction part of te formula is te slope of a secant line troug te points (x, f(x)) and (x+, f(x+)). Ten, as pictured on page 1 and defined above, we take te it as 0 to obtain te slope of a tangent line. (x+,f(x+)) (x,f(x)) m = y = x Exercise 2a: Use te algebraic definition of te derivative to determine te derivative function of f(x) = 3x 4. f (x) = 0 f ( x ) = Exercise 2b: Use te algebraic definition of te derivative to determine te derivative function of g(x) = x 2 x + 8. (See also Exercise 9 on page 9.) g (x) = 0 g( x ) g( x) =

4 Bob Brown Mat 251 Calculus 1 Capter 3, Section 1 Completed 4 Exercise 3: Use te algebraic definition of te derivative to determine te derivative function of f(x) = x 3 5x 2 4x Hint: (x + ) 3 = x 3 + 3x 2 + 3x f (x) = 0 f ( x ) = Exercise 4: Let g( x) x. Determine algebraically its derivative function. Compare te algebraic steps done ere to tose done in Exercise 4 on page 5 of Handout 2.3.

5 Bob Brown Mat 251 Calculus 1 Capter 3, Section 1 Completed 5 More Practice wit te Algebraic Definition of te Derivative Exercise 5a: Let f(x) = x 2. Determine Exercise 5b: Let f(x) = x 2. Determine f (x) algebraically. f (3) algebraically. Exercise 5c: Evaluate your answer for Exercise 5a at x = 3. Does tis matc your answer for Exercise 5b? Exercise 5d: Wat is te equation of te line tangent to te grap of f(x) = x 2 at te point were x = 3?

6 Bob Brown Mat 251 Calculus 1 Capter 3, Section 1 Completed 6 Wat te Derivative Function Tells Us About te Original Function Here is te actual derivative function from Exercise 1, neatened up a little bit. f (x) > 0 on te intervals f (x) < 0 on te interval f is increasing on te intervals f is decreasing on te interval General Principles If f (x) > 0 on an interval, ten f is on tat interval. If f (x) < 0 on an interval, ten f is on tat interval. Moreover, te magnitude of te derivative gives us te magnitude of te rate of cange. If f (x) is large (positive or negative), ten te grap of f is steep (up or down). If f (x) is small (positive or negative), ten te grap of f slopes gently (up or down). Wit tese facts in mind, we can deduce a lot about te beavior of a function from te beavior of its derivative function.

7 Bob Brown Mat 251 Calculus 1 Capter 3, Section 1 Completed 7 Te Derivative at a Point Does Not Always Exist Exercise 6a: Let x, if x 0 x. x, if x 0 Sow algebraically tat f does not ave a derivative at x = 0 by sowing tat te values of te one-sided its (in te it definition of te derivative) do not agree. Evaluate 0 f (0 ) f (0) Evaluate 0 f (0 ) f (0) Conclusion: Exercise 6b: Let x. Use an intuitive grapical approac (wic doesn t old te same weigt as an algebraic proof) tat f does not ave a derivative at x = 0.

8 Bob Brown Mat 251 Calculus 1 Capter 3, Section 1 Completed Exercise 7: Investigate te differentiability of x at x = 0. grapically f (0) = algebraically Exercise 8: Consider (i) Sketc y = g(x). x 3, if x 2 g ( x). 2x 1, if x 2 (ii) Is g continuous at x = 2? Prove your assertion. (iii) Is g differentiable at x = 2? Prove your assertion.

9 Bob Brown Mat 251 Calculus 1 Capter 3, Section 1 Completed 9 We see in Exercises 6, 7, and 8 tat altoug a function may be continuous at a point, it is not necessarily differentiable at tat point. We summarize tis in an important teorem. Teorem: If f is ten f is If f is ten f is Alternate Form of te Definition of te Derivative f (c) = xc f ( c) x c Exercise 9: Use te alternate form of te definition of te derivative to determine te derivative of te function g(x) = x 2 x + 8. (Recall Exercise 2b on page 3.)