By Samuel Horelick
This textbook is written for everybody who has skilled demanding situations studying Calculus. This ebook rather teaches you, is helping you already know and grasp Calculus via transparent and significant reasons of the entire principles, suggestions, difficulties and strategies of Calculus, powerful challenge fixing abilities and techniques, absolutely labored issues of whole, step by step motives.
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F(x) = x2 – x, at x = 2 4. f(x) = 9 – x2, at x = – 2 5. f(x) = 4x2 – 3, at (1, 1) 6. f(x) = 2x3 + 1, at x = – 1 7. f(x) = x2 + 3x – 4 at x = – 3 8. f(x) = – x3 + 2x + 4, at (0, 4) 9. f(x) = 1/(x + 3), at x = 3 10. f(x) = 2/(x + 3), at (1, ½) Average and Instantaneous Velocity: Velocity of an object is a rate of change of position with time. Average velocity describes how fast an object is moving during a given interval of time [t1, t2]. Average velocity is equal to distance traveled, which is D(t) = [D(t2) – D(t1)], divided by the time it took to travel this distance, that is (t2 – t1): V(average) = D/t = [D(t2) – D(t1)]/(t2 – t1 ) For example, the distance covered by a moving object is given by the formula D(t) = 3t + 5, where t is time in seconds.
Arithmetic combinations of Functions: Just as two numbers can be combined by operations of addition, subtraction, multiplication and division to form other numbers, two functions can be combined to create new functions. For example, f(x) = 2x – 3 and g(x) = x2 can be combined to form the sum, difference, product and quotient of f(x) and g(x): f(x) + g(x) = (2x – 3) + (x2) = x2 + 2x – 3 f(x) – g(x) = (2x – 3) – (x2) = – x2 + 2x – 3 f(x)g(x) = (2x – 3)(x2) = 2x3 – 3x2 f(x)/g(x) = (2x – 3)/(x2) = 2x/x2 – 3/x2 = 2/x – 3/x2, x ¹ 0 The domain of each new function will consist of all the numbers that the domains of f(x) and g(x) have in common.
The average velocity of this object between 6th and 8th second is [D(8) – D(6)] divided by (8 – 6), which is V(average) = [(3(8) + 5) – (3(6) + 5)] ¸ (8 – 6) = [29 – 23] ¸ 2 = 3. Instantaneous velocity is the speed of an object during an instant. Instantaneous velocity is the derivative of the distance function D(t) evaluated at an instant of time = t. For example, if the distance covered by a moving object is given by the formula D(t) = t2 + 7t, where t is time in seconds, then the instantaneous velocity of this object at 3rd second is the derivative of D(t), which is 2t + 7, evaluated at the 3rd second: 2t + 7 evaluated at t = 3 is 2(3) + 7 = 13.