Lesson 4 - Limits & Instantaneous Rates of Change

Transcription

1 Lesson Objectives Lesson 4 - Limits & Instantaneous Rates of Cange SL Topic 6 Calculus - Santowski 1. Calculate an instantaneous rate of cange using difference quotients and limits. Calculate instantaneous rates of cange numerically, grapically, and algebraically 3. Calculate instantaneous rates of cange in real world problems 4. Work wit tangent slopes & make te connection to instantaneous rates of cange 8/5/16 SL Topic 6 Calculus - Santowski 1 Fast Five 1. State te slope of te line perpendicular to te line x 3y = 6. Find te equation of a line passing troug (-3,5) and (4,-5) 3. A linear function is given by y = 1 x. If x increases by 4, by ow muc does y cange? 4. Find an expression for te slope of a line passing troug (x, f(x)) and (x+, f(x+)) (A) Opening Exercise: Average Rates of Cange: Motion If a ball is trown into te air wit a velocity of 30 m/s, its eigt,, in meters after t seconds is given by (t)=30t-5t. Determine te distance traveled in te 3 rd second. Find te average velocity for te time period beginning wen t = sec and lasting: " (i) 1 s " (ii) 0.5 s " (iii) 0.1 s So, now estimate te instantaneous velocity at t = 3 4 (B) Slopes of Tangents (B) Slopes of Tangents Now we want to take tis concept of rates of cange one step furter and develop a process wereby we can estimate instantaneous rates of cange and tangent line slopes Given te function f(x) = x, HOW WOULD YOU GRAPHICALLY (using TI-84) determine te slope of te tangent line to te curve at te point (1,1). Wat would your answer mean? 5 6 1

2 (B) Slopes of Tangents Grapically on TI-84 Here is ow we can visualize (and solve) te problem using te GDC # We simply ask te calculator to grap te tangent line (B) Slopes of Tangents # Meaning So finding te SLOPE OF A TANGENT line to a curve at a point is te same as finding te rate of cange of te curve at tat point We call tis te INSTANTANEOUS rate of cange of te function at te given point 7 8 (B) Slopes of Tangents Given te function f(x) = x, HOW WOULD YOU ALGEBRAICALLY determine te EXACT slope of te tangent line to te curve at te point (1,1). Wat would your answer mean? (B) Slopes of Tangents # Meaning Recall wat slopes mean # slope means a rate of cange (te cange in rise given te cange in run ) Recall te slope formulas: m = Δy Δx = y y 1 Δf (x) = = f ( x 1) f ( x ) x x 1 Δx x 1 x m = f ( x + Δx) f ( x) = f ( x + ) f ( x) Δx 9 10 (B) Estimating Tangent Slopes Using Secant Slopes For te time being we cannot directly use our algebra skills to find te slope of a line if we only know ONE point on te line STRATEGY: Let s use secant slopes and make estimates using a secant slope Given f(x) = x, use te TI-84 to estimate te slope of te tangent line to te curve at te point (1,1) If Q is a second point on te curve, find te slope of te secant PQ if te x-coordinate of Q is: " (i) x = (ii) x = 1.5 " (iii) x = 1.1 (iv) x = 1.01 " (v) x = (vi) x =

3 ttps:// bpiooejd8 If Q is a second point on te curve, find te slope of te secant PQ if te x-coordinate of Q is m = f (1+ ) f (1) (1+ ) (1) m = (1+ ) (1) ( ) ( 1) 1+ + m = m = + m = / ( + ) / m = Given f(x) = x, use te TI-84 to estimate te slope of te tangent line to te curve at te point (1,1) If te second point on te curve is at x = 1 +, te slope of te secant will be given m sec = + Recall te significance of Given f(x) = 4x x, we will estimate te slope of te tangent line to te curve at te point (1,3) If te second point on te curve is now 1 +, find te slope of te secant PQ Now predict te slope of te tangent line. So ow would we now estimate te slope of te tangent at x =? P 1, 1 lies on te curve If te second point on te curve is 1 +, find te slope of te secant PQ Now predict te slope of te line tangent to at x = 1 y = 1 x +1 y = 1 x +1 Given te function f(x) = x 3 + x, estimate te slope of te tangent line to te curve at te point (,8) If te second point on te curve is +, find te slope of te secant line and tus estimate te slope of te tangent line at x =

4 (E) Instantaneous Rates of Cange - Algebraic Calculation We will adjust te tangent slope formula as follows (to create a more algebra friendly formula) ( ) f ( x) f x + 0 (E) Instantaneous Rates of Cange - Algebraic Calculation Given te function f(x) = (x 1) 4, determine te slope of te curve at te point (,-3). Let Point Q be ( +, f( + )) Visualize tis process on DESMOS See te diagram on te next page for an explanation of te formula 19 0 (E) Instantaneous Rates of Cange - Algebraic Calculation (E) Example - Determine te Slope of a Curve f ( + ) f () 0 ( + ) () 0 (( + ) 1) 4) (( 1) 4) ((1+ ) 4) ( 3) 0 ( ) ( 3) 0 / ( + ) 0 / ( / + ) m = 0 8/5/16 1 Calculus - Santowski (E) Example - Determine te Slope of a Determine te slope of te tangent line to te curve f(x) = -x + 3x - 5 at te point (-4,-33) Alternate way to ask te same question: Determine te instantaneous rate of cange of f(x) = - x + 3x - 5 at te point (-4,-33) (E) Example - Determine te Slope of a f ( 4 + ) f ( 4) 0 ( 4 + ) + 3( 4 + ) 5 0 ( ) ( 33) ( ( 8 +16) + ( 1 + 3) 5) ( 33) 0 ( ) ( 33) 0 / ( +11) 0 / ( / +11) 0 m =

5 (E) Example - Determine te Slope of a Now let s confirm tis using te graping features and te calculus features of te TI-84 So ask te calculator to grap and determine te tangent eqn: (F) Furter Examples - Determine te Slope of a Make use of te limit definition of an instantaneous rate of cange to determine te instantaneous rate of cange of te following functions at te given x values: f (x) = x 5x + 40 at x = 4 g(x) =1 3x at x = 1 (x) = x 3 x at x =1 5 6 (G) Applications - Determine te Slope of a A football is kicked and its eigt is modeled by te equation (t) = -4.9t² + 16t + 1, were is eigt measured in meters and t is time in seconds. Determine te instantaneous rate of cange of eigt at 1 s, s, 3 s. (G) Applications - Determine te Slope of a Te solution is: m = d (1+ Δt) (1) = lim t=1 Δt ( ) ( 4.9( 1) +16( 1) +1) m = d 4.9( 1+ Δt) +16( 1+ Δt) +1 = lim t=1 Δt m = d ( 4.9Δt + 6.Δt +1.1) ( 1.1) = lim t=1 Δt m = d = lim( 4.9Δt + 6.) t=1 m = d dt t=1 = (G) Applications - Determine te Slope of a So te slope of te tangent line (or te instantaneous rate of cange of eigt) is 6. # so, in context, te rate of cange of a distance is called a speed (or velocity), wic in tis case would be 6. m/s at t = 1 sec. # now simply repeat, but use t =,3 rater tan 1 (G) Applications - Determine te Slope of a An object moves along a straigt line. Its position, x meters (from a fixed point O), at time, t seconds is given by x(t) = t 1 4 t, t 0 Determine te instantaneous velocity at t =

6 (G) Applications - Determine te Slope of a A business estimates its profit function by te formula P(x) = x 3 - x + were x is millions of units produced and P(x) is in billions of dollars. Determine te value of te tangent slope at x = ½ and at x = 1½. How would you interpret tese values (tat correspond to tangent slopes)? (G) Applications - Determine te Slope of a Te population of a city at te start of 000 was.3 million and its projected population, N million, is modeled by te equation given below, were t > 0 and is measured in years since te beginning of 000. N(t) =.3e 0.014t, t 0 " (a) Determine te average annual cange of te population. " (b) Use te TI-84 & idea of local linearity to estimate te growt of te city at te beginning of 005 (instantaneous RoC at t = 5) " (c) Use te TI-84 & draw te tangent line at t = 5 to determine te te growt of te city at te beginning of 005 " (d) Now try it algebraically..????