Math 2A Lecture 1D - Tangents & Limits PDF

# MATH 2A ## 2.1-2: Tangents and Limits ### Lecture 1D Assume that a car that constantly accelerates from rest has a position function *s(t) = t²*. * **Distance** * **Time** What is the average velocity of the car on the interval *[a, b]*. * **Average Speed:** *(s(b) - s(a)) / (b - a)...

# MATH 2A ## 2.1-2: Tangents and Limits ### Lecture 1D Assume that a car that constantly accelerates from rest has a position function *s(t) = t²*. * **Distance** * **Time** What is the average velocity of the car on the interval *[a, b]*. * **Average Speed:** *(s(b) - s(a)) / (b - a)* How can we approximate the exact velocity at time *t = 1*? * **[1, 2]:** *(s(2) - s(1)) / (2 - 1) = (4-1) / 1 = 3* * **[1, 1.1]:** *(s(1.1) - s(1)) / (1.1 - 1) = (1.21 - 1) / 0.1 = 2.1* * **[1, 1.5]:** *(s(1.5) - s(1)) / (1.5 - 1) = (2.25 - 1) / 0.5 = 2.5* * **[1, 1.01]:** *(s(1.01) - s(1)) / (1.01 - 1) = (1.0201 - 1) / 0.01 = 2.01* * **[1, 1 + h]:** *(s(1 + h) - s(1)) / (1 + h - 1) = (1 + 2h + h² - 1) / h = (2h + h²) / h = 2h / h = 2 + h* When we assume that *h* is very small, we are actually taking a limit. We write *lim f(x) = L* to mean when x is very close to a, *f(x)* gets very close to *L.* * **lim_h→0 (s(1 + h) - s(1))/h = 2** * **lim_x→1 2x+1 = 3** * **lim_x→-1 2x+1 = 3** * **lim_x→1- 2x+1 = 3** * **lim_x→1 x²= 1** * **lim_x→-1 x² = 1** * **lim_x→1 x²= 1** * **lim_x→2 (x-2)/(x²-4) = 1/4** * **lim_x→2 (x-2)/(x²-4) = 1/4** * **lim_x→2 (x-2)/(x²-4) = 1/4** * **lim_x→∞ 1/x² = +∞** * **lim_x→0 1/x² = +∞** * **lim_x→0- 1/x² = +∞** * **lim_x→0+ 1/x² = +∞** * **lim_x→0 |x| / x = DNE (Does Not Exist)** * **lim_x→0+ |x| / x = 1** * **lim_x→0- |x| / x = -1** These leads us to define **one-sided limits:** * **lim_x→a+ f(x) = L:** means when x is very close to a and x > a, then f(x) gets close to L. * **lim_x→a- f(x) = L:** means when x is very close to a and x < a, then f(x) gets close to L. Look back at our previous work to determine the following limits: * **lim_x→1+ 2x+1 = 3** * **lim_x→1- 2x+1 = 3** * **lim_x→1 2x+1 = 3** * **lim_x→1+ x² = 1** * **lim_x→1- x² = 1** * **lim_x→1 x² = 1** * **lim_x→2+ (x-2)/(x²-4) = 1/4** * **lim_x→2- (x-2)/(x²-4) = 1/4** * **lim_x→2 (x-2)/(x²-4) = 1/4** * **lim_x→0+ 1/x² = +∞** * **lim_x→0- 1/x² = +∞** * **lim_x→0 1/x² = +∞** * **lim_x→0+ |x| / x = 1** * **lim_x→0- |x| / x = -1** * **lim_x→0 |x| / x = DNE** So, under what conditions will a double-sided limit exist? * **If lim_x→a+ f(x) = L and lim_x→a- f(x) = L, then lim_x→a f(x) = L.** Do single-sided limits always exist? Try **lim_x→0+ sin (π/x)** * DNE * **x | sin(π/x)** * **0.1 | sin(10π) = 0** * **0.01 | sin(100π) = 0** * **7592 | sin(7592π) = 0** * **N | sin(Nπ) = 0** * **x | sin(π/x)** * **1/2 | sin(2π) = 1** * **1/3 | sin(3π) = 1** * **1/400 | sin(400π) = 1** * **1/(4N+1) | sin(4Nπ + π/2) = 1**