c ROB EBY Blinn College Mathematics Departm

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1 Section One The Basic Idea of a Limit Radar Gun Instantaneous Velocity Consider the position function s(t) = 4.9t 2 +30t+20 Complete the following table: Time Interval [, 2] [,.5] [,.] [,.0] [,.00] Average Velocity Determine the instantaneous velocity at θ = 0 for f(θ) = cos(/θ) Time Interval [, ] [ 0.5, 0.5] [ 0.2, 0.2] [ 0.0, 0.0] [ 0.00, 0.00] Average Velocity Interactive Figure llege Mathematics Departm

2 EBY Blinn College thematics Department Mathematics Department One very important idea for calculus is the idea of the it of a function. We will explore this idea graphically, numerically, and algebraically. On an intuitive level, the it is asking what value of the function should we have gotten. So we ask what is the it of f(x) as x goes to the value r. Notationally, we write this as: x r f(x) For the following function, fill in the table: f(r) x r r = 3 r = 0 r = 3 For the following function, fill in the table: h(r) x r r = 3 r = 0 r = 3 This gives rise to the idea of directional, or one sided its. We can ask what the it of f(x) is as x approaches a value from the left or from the right. Notationally, we write this as: x r f(x) x r + f(x) Foreachofthefollowingvaluesofrfindthefollowingits for the given function. f(r) x r x r + x r r = 3 r = 2 r = r = 0 r = r = 2 r = 3 llege Mathematics Departm

3 Section Three Techniques for Computing Limits There is also an algebraic side to its. For our class, they mostly consist of what are called removable discontinuities. The idea is if the it will not evaluate at first, do some algebra to find an equivalent it that will evaluate. Find the following its: x 2 4 x 3 x 2 x 2 4 x 2 x 2 x 2 4 x 2 x+2 x 4 x 4 x 2 The Derivative - radar gun f (x) df dx f(x+h) f(x) h 0 h Let N = f(t) be the total number of cans of New Republic consumed by Rob by age t, where t is in years. Interpret each of the following: f(4) = f (50) = 6 3 df dt (2) = 50 4 (f ) (450) = /70 llege Mathematics Departm

4 EBY Blinn College thematics Department Mathematics Department Use the definition to find the derivative of each function below: f(x) = 3x 6 h(r) = 3r 2 + The following its represent the slope of a curve y = f(x) at the points (r,f(r)). Determine a possible function and number, then calculate the it. (2+h) 4 6 h 0 h h 0 h+3 3 h Use the definition to find the derivative at the given value, then find the equation of the tangent line at that same value. f(x) = x at x = 0 llege Mathematics Departm g(t) = 3t 2,t =

5 EBY Blinn College thematics Department Mathematics Department The number of new subscriptions to a newspaper, y, in a month is a function of the amount, x, in dollars spent on advertising in that month, so y = f(x). Interpret the following statements: f(250) = 80 f (250) = 2 Use the statements given in part a to estimate: f(25) f(260) Finding the equation of the tangent line at t = for r(t) = 3t 2 6t+ Suppose we know that New Republic Brewing sold 27 barrels of beer in 204, and 329 barrels in 205. What is our best guess for the number of barrels they will sell in 2020? Now suppose we know that New Republic Brewing sold 27 barrels of beer in 204, and 329 barrels in 205. A guy in a Delorian drives up and say he knows that in 207 they sold 527 barrels and in barrels. What is our best guess for the number of barrels they sold in 206? llege Mathematics Departm

6 Squeeze Theorem Let E,B, and Y satisfy E(x) B(x) Y(x) for all values of x near r, except possibly at r. If E(x) = Y(x) = L, x r x r then B(x) = L x r Find the following it: x 0 x3 cos(2/x) another squeeze Prove that xcos(/x) = 0 x 0 llege Mathematics Departm

7 Special Limits for Trigonometric Functions sinx x 0 x = cosx = 0 x 0 x sin3x x 0 sin8x tan7x x 0 sin4x cos 2 θ θ 0 θ x 3 sin(x+3) x 2 +8x+5 θ 0 θ x 3 5 llege Mathematics Departm

8 EBY Blinn College thematics Department Mathematics Department Different Notations Graph the der. given the graph: f(x) = x 2 f(x) f(x) = x f(x) llege Mathematics Departm

9 Which is the function and which is the derivative, or is there no relationship? 4 y 6 y 4 y 3 2 x x 3 2 x From the graph of the function below, list the slopes in Increasing order: A C B D E A B C D E Explain what each of these its are finding. x 2 x+ 3 x 2 llege Mathematics Departm 3x 2 +4x 7 x x