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- 1. Review for Midterm II Math 1a December 2, 2007 Announcements Midterm II: Tues 12/4 7:00-9:00pm (SC Hall B) I have oﬃce hours Monday 1–2 and Tuesday 3–4 (SC 323) I’m aware of the missing audio on last week’s problem session video
- 2. Outline The Closed Interval Method Diﬀerentiation The First Derivative Test The Product Rule The Second Derivative Test The Quotient Rule Applications The Chain Rule Related Rates Implicit Diﬀerentiation Optimization Logarithmic Diﬀerentiation Miscellaneous The shape of curves Linear Approximation The Mean Value Theorem Limits of indeterminate The Extreme Value Theorem forms
- 3. Diﬀerentiation Learning Objectives state and use the product, use implicit diﬀerentiation quotient, and chain rules to ﬁnd the derivative of a diﬀerentiate all function deﬁned implicitly. “elementary” functions: use logarithic polynomials diﬀerentiation to ﬁnd the rational functions: quotients of polynomials derivative of a function root functions: rational powers given a function f and a trignometric functions: point a in the domain of a, sin/cos, tan/cot, sec/csc compute the linearization inverse trigonometric of f at a functions exponential and use a linear approximation logarithmic functions to estimate the value of a any composition of function functions like the above
- 4. The Product Rule Theorem (The Product Rule) Let u and v be diﬀerentiable at x. Then (uv ) (x) = u(x)v (x) + u (x)v (x)
- 5. The Quotient Rule Theorem (The Quotient Rule) Let u and v be diﬀerentiable at x, with v (x) = 0 Then u v − uv u (x) = v2 v
- 6. The Chain Rule Theorem (The Chain Rule) Let f and g be functions, with g diﬀerentiable at a and f diﬀerentiable at g (a). Then f ◦ g is diﬀerentiable at a and (f g ) (a) = f (g (a))g (a) ◦ In Leibnizian notation, let y = f (u) and u = g (x). Then dy dy du = dx du dx
- 7. Implicit Diﬀerentiation Any time a relation is given between x and y , we may diﬀerentiate y as a function of x even though it is not explicitly deﬁned.
- 8. Derivatives of Exponentials and Logarithms Fact dx a = (ln a)ax dx dx e = ex dx d 1 ln x = dx x d 1 loga x = dx (ln a)x
- 9. Logarithmic Diﬀerentiation If f involves products, quotients, and powers, then ln f involves it to sums, diﬀerences, and multiples
- 10. Outline The Closed Interval Method Diﬀerentiation The First Derivative Test The Product Rule The Second Derivative Test The Quotient Rule Applications The Chain Rule Related Rates Implicit Diﬀerentiation Optimization Logarithmic Diﬀerentiation Miscellaneous The shape of curves Linear Approximation The Mean Value Theorem Limits of indeterminate The Extreme Value Theorem forms
- 11. The shape of curves Learning Objectives use the Closed Interval Test to classify critical Method to ﬁnd the points as relative maxima, maximum and minimum relative minima, or neither. values of a diﬀerentiable given a function, graph it function on a closed completely, indicating interval zeroes (if they are easily state Fermat’s Theorem, found) asymptotes (if the Extreme Value applicable) Theorem, and the Mean critical points Value Theorem relative/absolute use the First Derivative max/min inﬂection points Test and Second Derivative
- 12. The Mean Value Theorem Theorem (The Mean Value Theorem) Let f be continuous on [a, b] and diﬀerentiable on (a, b). Then there exists a point c in (a, b) such that • b f (b) − f (a) • = f (c). a b−a
- 13. The Mean Value Theorem Theorem (The Mean Value Theorem) Let f be continuous on [a, b] and diﬀerentiable on (a, b). Then there exists a point c in (a, b) such that • b f (b) − f (a) • = f (c). a b−a
- 14. The Mean Value Theorem Theorem (The Mean Value c Theorem) • Let f be continuous on [a, b] and diﬀerentiable on (a, b). Then there exists a point c in (a, b) such that • b f (b) − f (a) • = f (c). a b−a
- 15. The Extreme Value Theorem Theorem (The Extreme Value Theorem) Let f be a function which is continuous on the closed interval [a, b]. Then f attains an absolute maximum value f (c) and an absolute minimum value f (d) at numbers c and d in [a, b].
- 16. The Closed Interval Method Let f be a continuous function deﬁned on a closed interval [a, b]. We are in search of its global maximum, call it c. Then: This means to ﬁnd the Either the maximum maximum value of f on [a, b], occurs at an endpoint of we need to check: the interval, i.e., c = a a and b or c = b, Points x where f (x) = 0 Or the maximum occurs inside (a, b). In this case, Points x where f is not c is also a local diﬀerentiable. maximum. The latter two are both called Either f is critical points of f . This diﬀerentiable at c, in technique is called the Closed which case f (c) = 0 Interval Method. by Fermat’s Theorem. Or f is not diﬀerentiable at c.
- 17. The First Derivative Test Let f be continuous on [a, b] and c in (a, b) a critical point of f . Theorem If f (x) > 0 on (a, c) and f (x) < 0 on (c, b), then f (c) is a local maximum. If f (x) < 0 on (a, c) and f (x) > 0 on (c, b), then f (c) is a local minimum. If f (x) has the same sign on (a, c) and (c, b), then (c) is not a local extremum.
- 18. The Second Derivative Test Let f , f , and f be continuous on [a, b] and c in (a, b) a critical point of f . Theorem If f (c) < 0, then f (c) is a local maximum. If f (c) > 0, then f (c) is a local minimum. If f (c) = 0, the second derivative is inconclusive (this does not mean c is neither; we just don’t know yet).
- 19. Outline The Closed Interval Method Diﬀerentiation The First Derivative Test The Product Rule The Second Derivative Test The Quotient Rule Applications The Chain Rule Related Rates Implicit Diﬀerentiation Optimization Logarithmic Diﬀerentiation Miscellaneous The shape of curves Linear Approximation The Mean Value Theorem Limits of indeterminate The Extreme Value Theorem forms
- 20. Applications Learning Objectives model word problems with mathematical functions (this is a major goal of the course!) apply the chain rule to mathematical models to relate rates of change use optimization techniques in word problems
- 21. Related Rates Strategies for Related Rates Problems 1. Read the problem. 2. Draw a diagram. 3. Introduce notation. Give symbols to all quantities that are functions of time (and maybe some constants) 4. Express the given information and the required rate in terms of derivatives 5. Write an equation that relates the various quantities of the problem. If necessary, use the geometry of the situation to eliminate all but one of the variables. 6. Use the Chain Rule to diﬀerentiate both sides with respect to t. 7. Substitute the given information into the resulting equation and solve for the unknown rate.
- 22. Optimization Strategies for Optimization Problems 1. Read the problem. 2. Draw a diagram. 3. Introduce notation. Give symbols to all quantities that are functions of time (and maybe some constants) 4. Write an equation that relates the various quantities of the problem. If necessary, use the geometry of the situation to eliminate all but one of the variables. 5. Use either the Closed Interval Method, the First Derivative Test, or the Second Derivative Test to ﬁnd the maximum value of this function
- 23. Outline The Closed Interval Method Diﬀerentiation The First Derivative Test The Product Rule The Second Derivative Test The Quotient Rule Applications The Chain Rule Related Rates Implicit Diﬀerentiation Optimization Logarithmic Diﬀerentiation Miscellaneous The shape of curves Linear Approximation The Mean Value Theorem Limits of indeterminate The Extreme Value Theorem forms
- 24. Linear Approximation Let f be diﬀerentiable at a. What linear function best approximates f near a?
- 25. Linear Approximation Let f be diﬀerentiable at a. What linear function best approximates f near a? The tangent line, of course!
- 26. Linear Approximation Let f be diﬀerentiable at a. What linear function best approximates f near a? The tangent line, of course! What is the equation for the line tangent to y = f (x) at (a, f (a))?
- 27. Linear Approximation Let f be diﬀerentiable at a. What linear function best approximates f near a? The tangent line, of course! What is the equation for the line tangent to y = f (x) at (a, f (a))? L(x) = f (a) + f (a)(x − a)
- 28. Theorem (L’Hˆpital’s Rule) o Suppose f and g are diﬀerentiable functions and g (x) = 0 near a (except possibly at a). Suppose that lim f (x) = 0 and lim g (x) = 0 x→a x→a or lim f (x) = ±∞ lim g (x) = ±∞ and x→a x→a Then f (x) f (x) lim = lim , x→a g (x) x→a g (x) if the limit on the right-hand side is ﬁnite, ∞, or −∞.
- 29. Theorem (L’Hˆpital’s Rule) o Suppose f and g are diﬀerentiable functions and g (x) = 0 near a (except possibly at a). Suppose that lim f (x) = 0 and lim g (x) = 0 x→a x→a or lim f (x) = ±∞ lim g (x) = ±∞ and x→a x→a Then f (x) f (x) lim = lim , x→a g (x) x→a g (x) if the limit on the right-hand side is ﬁnite, ∞, or −∞. ∞ L’Hˆpital’s Rule also applies for limits of the form o . ∞
- 30. Cheat Sheet for L’Hˆpital’s Rule o Form Method 0 L’Hˆpital’s rule directly o 0 ∞ L’Hˆpital’s rule directly o ∞ ∞ 0 0·∞ jiggle to make or ∞. 0 ∞−∞ factor to make an indeterminate product 00 take ln to make an indeterminate product ∞0 ditto 1∞ ditto

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