Slie 1 Calculus I Announcements Office Hours: Amos Eaton 309, Monays 12:50-2:50 Exam 2 is Thursay, October 22n. The stuy guie is now on the course web page. Start stuying now, an make a plan to succee. Rea sections 3.2,3.3, an 3.6. (We are skipping 3.4, but we will cover 3.5 next lecture.) Do the homework for sections as follows 3.2, (Note that in Section 3.2, you nee to use the efinition on some of the questions, as per the instructions.) 3.3, o all of it 3.6, o up through 31 on the stuy guie, the last 4 questions on the stuy guie from 3.6 can be postpone until after Thursay s lecture.
Slie 2 Derivatives, Overview It is important to know at least 3 aspects of the erivative: The Definition Recall that the efinition of erivative of a function f with respect to a variable x is f f(x + h) f(x) (x) = lim h 0 h The Rules Since erivatives are use so often, rules are create to make them easier to fin. Here is a list of names of a few of the rules we will learn Power rule, sum rule, constant multiple rule, prouct rule, quotient rule, rule for exponentials, rules for trigonometric functions, chain rule, rules for logarithms, rules for inverse functions. Interpretations/Applications The erivative of a function gives the slope of the tangent line to the function. The erivative of a function gives the (instantaneous) rate of change of the function. The erivative of a population with respect to time gives the rate of change of the population. The erivative of position with respect to time gives instantaneous velocity The erivative of velocity with respect to time gives acceleration The erivative of mass with respect to istance gives linear ensity The erivative of charge with respect to time gives current Many, many other applications...
Slie 3 Derivatives Likely Exam Essay Question: State the efinition of erivative. Illustrate the efinition with a picture an explain how the erivative gives the instantaneous slope of the graph.
Slie 4 Derivatives: Notation You nee to know ifferent notations for the erivative of a function y = f(x). f (x) = y = y x = f x = x f(x) = D(f)(x) = D xf(x) All of these are equal to f (x) = lim h 0 f(x + h) f(x) h Also, for the erivative of the function f(x) at a particular point x = a, we use all the following notations f (a) = y x = f x=a x = x=a x f(x) x=a
Slie 5 Sketching Derivatives The erivative of a function is another function. 1. Piecewise linear functions The erivative of a straight line y = mx + b is a constant function y = m Given the graph of a piecewise linear function, sketch the graph of the erivative. 2. Smooth functions The erivative of a general function gives its slope at each point (i.e. the slope of the tangent line). Given the graph of a general function, sketch the graph of the erivative.
Slie 6 Differentiability f(a + h) f(a) A function is ifferentiable at x = a if lim h 0 h exists. If a function is ifferentiable at x = a then the function is continuous at x = a But the previous statement is a one-way-street: a function can be continuous at x = a, but not ifferentiable at x = a. Draw a graph to illustrate this. For the graph of a given function, ientify where it appears to be iscontinuous an where it appears to be non-ifferentiable.
Slie 7 Derivative Rules Recall f (x) = lim h 0 f(x + h) f(x) h 1. Constant Rule c = 0 when c is constant x 2. Power Rule x xn = nx n 1 (for any real number n 0) 3. Constant multiple rule x cf(x) = cf (x) 4. Sum rule x (f(x) + g(x)) = f (x) + g (x) Use the rules to fin ( x 4 + 3x 2 x ) x Use the rules to fin x 5 + x 2 x x 3 Derive the constant rule Derive the power rule for integers n = 2, 3
Slie 8 iclicker Question Fin y if y = 4x 3 + 3x + ln(7) A. 12x 2 + 3 + 1/7 B. 12x 2 + 3 C. 4x 2 + 3 + 1/7 D. 4x 2 + 3 E. None of the above
Answer to Question Fin y if y = 4x 3 + 3x + ln(7) A. 12x 2 + 3 + 1/7 B. 12x 2 + 3 is the correct answer. C. 4x 2 + 3 + 1/7 D. 4x 2 + 3 E. None of the above
Slie 9 Derivative Rules Here s a list of a few of the stanar erivatives to memorize. We will erive some of these later. 1. c = 0 for any constant c x 2. x x = 1 3. x xr = rx r 1 for r 0 4. x ex = e x 5. x ax = a x ln(a) for a > 0 6. ln(x) = 1/x x 7. sin(x) = cos(x) x 8. cos(x) = sin(x) x 9. x tan(x) = sec2 (x) 10. x cot(x) = csc2 (x) 11. sec(x) = sec(x) tan(x) x 12. csc(x) = csc(x) cot(x) x
Slie 10 iclicker Question Fin f (x) if f(x) = 3 x A. 3 x+1 /(x + 1) B. x3 x 1 C. 3 x D. 3 x ln(3) E. None of the above
Answer to Question Fin f (x) if f(x) = 3 x A. 3 x+1 /(x + 1) B. x3 x 1 C. 3 x D. 3 x ln(3) is the correct answer. E. None of the above
Slie 11 iclicker Question Fin f (x) if f(x) = sec(x) + cot(x) A. tan 2 (x) sec 2 (x) B. tan 2 (x) csc 2 (x) C. sec(x) tan(x) sec 2 (x) D. sec(x) tan(x) csc 2 (x) E. None of the above
Answer to Question f(x) = sec(x) + cot(x) Fin f (x) if A. tan 2 (x) sec 2 (x) B. tan 2 (x) csc 2 (x) C. sec(x) tan(x) sec 2 (x) D. sec(x) tan(x) csc 2 (x) is the correct answer. E. None of the above
Slie 12 Derivative Rules If f an g are ifferentiable functions, the Prouct Rule for u = f(x) an v = g(x) says or x (f(x) g(x)) = f (x)g(x) + f(x)g (x) u uv = x x v + uv x 1. Use the prouct rule to fin the following erivatives: x (x2 + 3x + 5) (x 5 3x + 2) x ex sin(x) x (x3 + 3x + 4) cos(x) 2. Derive the prouct rule (Possible lecture question on exam 2, will be one in class on October 1.)
Slie 13 iclicker Question Fin x ln(x)(x5 + 3x 2) A. 5x4 + 3 x B. ln(x)(5x 4 + 3) C. ln(x)(5x 4 + 3) + x5 + 3x 2 x D. ln(x)(x 5 + 3x 2) + 5x4 + 3 x E. None of the above
Answer to Question Fin x ln(x)(x5 + 3x 2) A. 5x4 + 3 x B. ln(x)(5x 4 + 3) C. ln(x)(5x 4 + 3) + x5 + 3x 2 x answer. D. ln(x)(x 5 + 3x 2) + 5x4 + 3 x E. None of the above is the correct
Slie 14 Derivative Rules For ifferentiable functions f an g, (g(x) 0) the Reciprocal Rule says ( ) 1 = g (x) x g(x) (g(x)) 2 an the Quotient Rule says ( ) f(x) = g(x)f (x) f(x)g (x) x g(x) (g(x)) 2 1. Use the rules to fin x x 1 sin(x) x 2 + 3x + 2 e x + cos(x)
Slie 15 iclicker Question Fin x 2 + 3x x 4x + 5 A. 4(x2 + 3x) (2x + 3)(4x + 5) (4x + 5) 2 B. (4x + 5)(x2 + 3x) 4(2x + 3) (4x + 5) 2 C. (2x + 3)4 (4x + 5)(x2 + 3x) (4x + 5) 2 D. (2x + 3)(4x + 5) 4(x2 + 3x) (4x + 5) 2 E. None of the above
Answer to Question Fin x 2 + 3x x 4x + 5 A. 4(x2 + 3x) (2x + 3)(4x + 5) (4x + 5) 2 B. (4x + 5)(x2 + 3x) 4(2x + 3) (4x + 5) 2 C. (2x + 3)4 (4x + 5)(x2 + 3x) (4x + 5) 2 D. (2x + 3)(4x + 5) 4(x2 + 3x) (4x + 5) 2 is the correct answer. E. None of the above