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- Title: ho26
- Type: Notes
- School: Rutgers
- Course: CALC 1
- Term: Fall
Coursehero >> New Jersey >> Rutgers >> CALC 1
Path: Rutgers >> CALC >> 1 Fall, 2008
Path: Rutgers >> CALC >> 1 Fall, 2008
Path: Rutgers >> CALC >> 1 Fall, 2008
Path: Rutgers >> CALC >> 1 Fall, 2008
Path: Rutgers >> CALC >> 1 Fall, 2008
Path: Rutgers >> CALC >> 1 Fall, 2008
Path: Rutgers >> CALC >> 1 Fall, 2008
Path: Rutgers >> CALC >> 1 Fall, 2008
Path: Rutgers >> CALC >> 1 Fall, 2008
Path: Rutgers >> CALC >> 1 Fall, 2008
Path: Rutgers >> CALC >> 1 Fall, 2008
Path: Rutgers >> CALC >> 1 Fall, 2008
Path: Rutgers >> CALC >> 1 Fall, 2008
Path: Rutgers >> CALC >> 1 Fall, 2008
Path: Rutgers >> CALC >> 1 Fall, 2008
Path: Rutgers >> CALC >> 1 Fall, 2008
Path: Rutgers >> CALC >> 1 Fall, 2008
Path: Rutgers >> CALC >> 1 Fall, 2008
Path: Rutgers >> CALC >> 1 Fall, 2008
Path: Rutgers >> CALC >> 1 Fall, 2008
Path: Rutgers >> CALC >> 1 Fall, 2008
Path: Rutgers >> CALC >> 1 Fall, 2008
Path: Rutgers >> CALC >> 1 Fall, 2008
Path: Rutgers >> CALC >> 1 Fall, 2008
Path: Rutgers >> CALC >> 1 Fall, 2008
Path: Rutgers >> CALC >> 1 Fall, 2008
Path: Rutgers >> CALC >> 1 Fall, 2008
Path: Rutgers >> CALC >> 1 Fall, 2008
Path: Rutgers >> CALC >> 1 Fall, 2008
Path: Rutgers >> CALC >> 1 Fall, 2008
Path: Rutgers >> CALC >> 1 Fall, 2008
Path: Rutgers >> CALC >> 1 Fall, 2008
Path: Rutgers >> CALC >> 1 Fall, 2008
Path: Rutgers >> CALC >> 1 Fall, 2008
Path: Rutgers >> CALC >> 1 Fall, 2008
Path: Rutgers >> CALC >> 1 Fall, 2008
Path: Rutgers >> CALC >> 1 Fall, 2008
Path: Rutgers >> CALC >> 1 Fall, 2008
Path: Rutgers >> CALC >> 1 Fall, 2008
Path: Rutgers >> CALC >> 1 Fall, 2008
Path: Rutgers >> CALC >> 1 Fall, 2008
Path: Rutgers >> CALC >> 1 Fall, 2008
Path: Rutgers >> CALC >> 1 Fall, 2008
Path: Rutgers >> CALC >> 1 Fall, 2008
Path: Rutgers >> CALC >> 1 Fall, 2008
Path: Rutgers >> CALC >> 1 Fall, 2008
Path: Rutgers >> CALC >> 1 Fall, 2008
Path: Rutgers >> CALC >> 1 Fall, 2008
Path: Rutgers >> CALC >> 1 Fall, 2008
Path: Rutgers >> CALC >> 1 Fall, 2008
Path: Rutgers >> CALC >> 1 Fall, 2008
Path: Rutgers >> CALC >> 1 Fall, 2008
Path: Rutgers >> CALC >> 1 Fall, 2008
Path: Rutgers >> CALC >> 1 Fall, 2008
Path: Rutgers >> CALC >> 1 Fall, 2008
Path: Rutgers >> CALC >> 1 Fall, 2008
Path: Rutgers >> CALC >> 1 Fall, 2008
Path: Rutgers >> CALC >> 1 Fall, 2008
Path: Rutgers >> CALC >> 1 Fall, 2008
Path: Rutgers >> CALC >> 1 Fall, 2008
Path: Rutgers >> CALC >> 1 Fall, 2008
Path: Rutgers >> CALC >> 1 Fall, 2008
Path: Rutgers >> CALC >> 1 Fall, 2008
Path: Rutgers >> CALC >> 1 Fall, 2008
Path: Rutgers >> CALC >> 1 Fall, 2008
Path: Rutgers >> CALC >> 1 Fall, 2008
Path: Rutgers >> CALC >> 1 Fall, 2008
Path: Rutgers >> CALC >> 1 Fall, 2008
Path: Rutgers >> CALC >> 1 Fall, 2008
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Z Dr. s Math151 Handout # 2.6 [Trigonometric Limits] By Doron Zeilberger Problem Type 2.6.1: Use the Squeeze Theorem to prove that limx a GoesT oZero(x) Bounded(x) = 0 Example Problem 2.6.1 : Use the Squeeze Theorem to prove that lim x3 + x2 sin( /x) = 0 x 0 Steps 1. Show that lim GoesT oZero(x) Example 1. x 0 indeed equals 0. x 0 lim x3 + x2 = 03 + 02 = 0 = 0. 2. Show that Bounded(x) is indeed bounded, 2. The sin function is always between 1 at least near x = a. and 1 hence is always bounded. Problem Type 2.6.2: Evaluate t 0 lim ExpressionInSinesAndOrCosines(t) Example Problem 2.6.2 : Evaluate lim sin 7x 3x . x 0 Steps 1. The o cial way is to rearrange things in such a way that you would be able to use one (or both) of the following lim sin w =1 . w Example 1. in order to accomodate the sin 7x we divide and multiply by 7x: lim sin 7x 3x (sin 7x)(7x) (7x)(3x) x 0 x 0 w 0 w 0 lim 1 cos w =0 , w = = lim lim x 0 where w is whatever that goes to 0. x 0 sin 7x 7x lim lim 7x 3x = 7x sin 0 7x 7x 7 7 = 3 3 . 7 = 3 =1 Ans.: 7 3 . 2. The uno cial (shortcut) way is to replace sin w by w everywhere. 2. sin 7x x 0 3x 7x 7 7 = lim = lim = x 0 3x x 0 3 3 lim . Problem Type 2.6.3: Evaluate t 0 lim ExpressionInSinesAndOrCosines(t) Example Problem 2.6.3 : Evaluate lim cos 5x cos 3x x . x 0 Steps 1. Use algebra to rearrange the top so that you would be able to take advantage of 1 cos w lim =0 , w 0 w where w is whatever that goes to 0. Example 1. x 0 lim cos 5x cos 3x x = lim = lim x 0 (1 cos 5x) (1 cos 3x) x x 0 (1 cos 5x) (1 cos 3x) lim x 0 x x (1 cos 5x) (1 cos 3x) 3 lim x 0 5x 3x (1 cos 5x) (1 cos 3x) 3 lim = 0 0 = 0 3x 0 5x 3x = 5 lim x 0 = 5 lim 5x 0 Ans.: 0. Note: Much later in this semester, you will learn a better way of doing these, called L H pital s o rule, but until then you have to do it today s way. In particular, in Midterm One you can t use L H pital s rule, in case you are familiar with it. o
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ho27.pdf
Path: Rutgers >> CALC >> 1 Fall, 2008
Description: Dr. Zs Math151 Handout #2.7 [The Intermediate Value Theorem] By Doron Zeilberger Problem Type 2.7.1 : If f (x) = Expression(x), show that there is a number c such that f (c) = N umber. Example Problem 2.7.1: If f (x) = x3 x2 + 3x, show that there is...
ho28.pdfPath: Rutgers >> CALC >> 1 Fall, 2008
Description: Dr. Zs Math151 Handout #2.8 [The Formal Denition of a Limit] By Doron Zeilberger Problem Type 2.8.1 : Prove rigorously that xc lim f (x) = L , for some numbers c and L and some function (usually very simple) function f (x). Example Problem 2.8.1: ...
ho31.pdfPath: Rutgers >> CALC >> 1 Fall, 2008
Description: Dr. Zs Math151 Handout #3.1 Problem Type 3.1.1: Compute the derivative at the ven point x = a, of a given function f (x), using the limit denition and nd an equation of the tangent line. Example Problem 3.1.1: Compute the derivative at x = 2, if f (x...
ho32.pdfPath: Rutgers >> CALC >> 1 Fall, 2008
Description: Dr. Zs Math151 Handout #3.2 Problem Type 3.2.1: Compute f (x) from the limit denition (no credit for other methods!), where f (x) is as given Example Problem 3.2.1: Compute f (x) from the limit denition (no credit for other methods!), where f (x) = x...
ho33.pdfPath: Rutgers >> CALC >> 1 Fall, 2008
Description: Dr. Zs Math151 Handout #3.3 [The Product and Quotient Rule] By Doron Zeilberger Problem Type 3.3.1 : Dierentiate f (x) = Expression1 (x) Expression2 (x), where both expressions are easy to dierentiate from known rules. Example Problem 3.3.1: Dierent...
ho34.pdfPath: Rutgers >> CALC >> 1 Fall, 2008
Description: Dr. Zs Math151 Handout #3.4 [Rates of Change] By Doron Zeilberger Problem Type 3.4.1 : A particle moves according to a law of motion s = f (t), t > 0, where t is measured in some unit of time and s in some unit of distance. (a) Find the velocity at a...
ho35.pdfPath: Rutgers >> CALC >> 1 Fall, 2008
Description: Dr. Zs Math151 Handout #3.5 [Higher Derivatives] By Doron Zeilberger Problem Type 3.5.1 : Find the rst, second, and third derivatives of the function f (x) = Expression(x). Example Problem 3.5.1: Find the rst, second, and third derivatives of the fun...
ho36.pdfPath: Rutgers >> CALC >> 1 Fall, 2008
Description: Dr. Zs Math151 Handout #3.6 [Trigonometric Functions] By Doron Zeilberger Problem Type 3.6.1 : Dierentiate an expression involving products and/or quotients of expressions containing trig functions. Example Problem 3.6.1: Dierentiate y= 1 + cos x x +...
ho37.pdfPath: Rutgers >> CALC >> 1 Fall, 2008
Description: Dr. Zs Math151 Handout #3.7 [The Chain Rule] By Doron Zeilberger Problem Type 3.7.1 : Write the composite function in the form f (g(x), Identify the inner function u = g(x) and the outer function y = f (u). Then nd the derivative dy/dx. Example Probl...
ho38.pdfPath: Rutgers >> CALC >> 1 Fall, 2008
Description: Dr. Zs Math151 Handout #3.8 [Implicit Dierentiation] By Doron Zeilberger Problem Type 3.8.1 : Find dy/dx by implicit dierentiation, where you know that Expression1 (x, y) = Expression2 (x, y). Example Problem 3.8.1: Find dy/dx by implicit dierentiati...
ho39.pdfPath: Rutgers >> CALC >> 1 Fall, 2008
Description: Dr. Zs Math151 Handout #3.9 [Derivatives of Inverse Functions] By Doron Zeilberger Problem Type 3.9.1 : Use the formula g (x) = 1 f (g(x) , to calculate g (x), where g(x) is the inverse of the given function f (x). Example Problem 3.9.2: Use the ...
ho310.pdfPath: Rutgers >> CALC >> 1 Fall, 2008
Description: Dr. Zs Math151 Handout #3.10 [Derivatives of General Exponential and Logarithmic Functions] By Doron Zeilberger Problem Type 3.10.1 : Dierentiate the function f (x) = Expression(x), where the expression involves ln x or loga x. Example Problem 3.10.1...
ho311.pdfPath: Rutgers >> CALC >> 1 Fall, 2008
Description: Dr. Zs Math151 Handout #3.11 [Related Rates] By Doron Zeilberger Problem Type 3.11.1 : If F (x, y) = c and dy/dt = a, nd dx/dt when y = b. Example Problem 3.11.1: If x3 + y 3 = 9 and dy/dt = 6 nd dx/dt when y = 2. Steps 1. Find the corresponding val...
ho41.pdfPath: Rutgers >> CALC >> 1 Fall, 2008
Description: Dr. Zs Math151 Handout #4.1 Problem Type 4.1.1: Use the Linear Approximation to estimate f = f (a + h) f (a) for the given function f (x), for the given a and h. Example Problem 4.1.1: Use the Linear Approximation to estimate f = f (4 + 0.01) f (4)...
ho42.pdfPath: Rutgers >> CALC >> 1 Fall, 2008
Description: Dr. Zs Math151 Handout #4.2 [Extreme Values] By Doron Zeilberger Problem Type 4.2.1 : Find the critical numbers of the function f (x) = Expression(x). Example Problem 4.2.1: Find the critical numbers of the function f (x) = x3 + 3x2 24x. Steps 1. Fi...
ho43.pdfPath: Rutgers >> CALC >> 1 Fall, 2008
Description: Dr. Zs Math151 Handout #4.3 [The Mean Value Theorem and Monotinicity] By Doron Zeilberger Problem Type 4.3.1 : Verify that the function f (x) satises the hypothesis of the Mean Value Theorem on the interval [a, b]. Then nd all numbers c that satisfy ...
ho44.pdfPath: Rutgers >> CALC >> 1 Fall, 2008
Description: Dr. Zs Math151 Handout #4.4 [The Shape of a Graph] By Doron Zeilberger Problem Type 4.4.1 : Given a function f (x) = P olynomial(x), (a) Find the intervals on which f is increasing or decreasing. (b) Find the local maximum and minimum values of f . (...
ho45.pdfPath: Rutgers >> CALC >> 1 Fall, 2008
Description: Dr. Zs Math151 Handout #4.5 [Graph Sketching and Asymptotes] By Doron Zeilberger Problem Type 4.5.1 : Sketch the curve y = P olynomial(x). Example Problem 4.5.1: Sketch the curve y = x4 + 4x3 Steps 1. Find the rst and second derivatives dy/dx and d2...
ho46.pdfPath: Rutgers >> CALC >> 1 Fall, 2008
Description: Dr. Zs Math151 Handout #4.6 [Applied Optimization] By Doron Zeilberger Problem Type 4.6.1 : A farmer wants to fence an area of A square units and then divide it into n + 1 parts by placing n parallel fences parallel to one of the sides of the rectang...
ho47.pdfPath: Rutgers >> CALC >> 1 Fall, 2008
Description: Dr. Zs Math151 Handout #4.7 [LHspitals Rule] o By Doron Zeilberger Problem Type 4.7.1 : Given certain limits of certain functions, f (x), g(x), . . . at a designated point x = a, determine whether the limits (at that very same point x = a) of the quo...
ho48.pdfPath: Rutgers >> CALC >> 1 Fall, 2008
Description: Dr. Zs Math151 Handout #4.8 [Newtons Method] By Doron Zeilberger Problem Type 4.8.1 : Use Newtons method with the specied initial approximation x1 to nd x3 , the third approximation to the root of the given equation. (Give your answer to four decimal...
ho49.pdfPath: Rutgers >> CALC >> 1 Fall, 2008
Description: Dr. Zs Math151 Handout #4.9 [Antiderivatives] By Doron Zeilberger Problem Type 4.9.1 : Find the most general antiderivative of the function f (x). Example Problem 4.9.1: Find the most general antiderivative of the function f (x) = 5ex + 8 sec2 x. Ste...
ho51.pdfPath: Rutgers >> CALC >> 1 Fall, 2008
Description: Dr. Zs Math151 Handout #5.1 [The Denite Integral] By Doron Zeilberger Problem Type 5.1.1 : Use the denition of the integral b n f (x)dx = lim a n f (xi )x i=1 (where x = (b a)/n and xi = a + ix). to evaluate the integral b f (x)dx a Example Pr...
ho52.pdfPath: Rutgers >> CALC >> 1 Fall, 2008
Description: Dr. Zs Math151 Handout #5.2 Problem Type 5.2.1: Draw a graph of the signed area represented by the integral b f (x) dx a and compute it using geometry. Example Problem 5.2.1: Draw a graph of the signed area represented by the integral 1 (3x + 4) d...
ho53.pdfPath: Rutgers >> CALC >> 1 Fall, 2008
Description: Dr. Zs Math151 Handout #5.3 [The Fundamental Theorem of Calculus, Part I] By Doron Zeilberger Problem Type 5.3.1 : Evaluate the (denite) integral b f (V ar)d V ar a Example Problem 5.3.1: Evaluate the (denite) integral 1 (u5 u3 + u2 )d u 1 Steps...
ho54.pdfPath: Rutgers >> CALC >> 1 Fall, 2008
Description: Dr. Zs Math151 Handout #5.4 [The Fundamental Theorem of Calculus, Part II] By Doron Zeilberger Problem Type 5.4.1 : Dierentiate V ariable1 f (V ariable1 ) = N umber Expression(V ariable2 )d V ariable2 . Example Problem 5.4.1: Use the Fundamental ...
ho56.pdfPath: Rutgers >> CALC >> 1 Fall, 2008
Description: Dr. Zs Math151 Handout #5.6 [The Substitution Rule] By Doron Zeilberger Problem Type 5.6.1 : Evaluate the indenite integral COM P LICAT ED(V ar) dV ar . Example Problem 5.6.1: Evaluate ex 1 + ex dx . Steps 1. Try to nd a good u. Usually whats insi...
qSept4.pdfPath: Rutgers >> CALC >> 1 Fall, 2008
Description: QUIZ for Sept. 4, 2008 NAME: (print!) Section: E-MAIL ADDRESS: (print!) 1. What are: (a) sin 0 (b) cos 0 (c) ln 1 . 2. What is the equation of the line joining (1, 1) and (2, 3)? 3. If f (x) = 1 x+1 , what is the inverse function f 1 (x)?. ...
qASept4.pdfPath: Rutgers >> CALC >> 1 Fall, 2008
Description: Solutions to QUIZ for Sept. 4, 2008 1. What are: (a) sin 0 Sol. to 1. (a) sin 0 = 0 (b) cos 0 = 1 (c) ln 1 = 0. Common Mistakes: Everyone got (a) and (b) right, but quite a few people messed up (c). Common wrong answers were 1 and e. It is true that ...
qSept8.pdfPath: Rutgers >> CALC >> 1 Fall, 2008
Description: QUIZ for Monday, Sept. 8, 2008 NAME: (print!) Section: E-MAIL ADDRESS: (print!) 1. : (a) Draw the function x + 2, 7, f (x) = 2x + 2, 5, 3x 2, if if if if if x < 1; x = 1; 1 < x < 2; x = 2; x > 2. (b) What are: x1 lim f (x) = x1+ li...
qASept8.pdfPath: Rutgers >> CALC >> 1 Fall, 2008
Description: Solution to the QUIZ for Monday, Sept. 8, 2008 1. : (a) Draw the function x + 2, 7, f (x) = 2x + 2, 5, 3x 2, if if if if if x < 1; x = 1; 1 < x < 2; x = 2; x > 2. Solution. I am too lazy to use a graphing program to plot it, but let me tel...
qSept11.pdfPath: Rutgers >> CALC >> 1 Fall, 2008
Description: QUIZ for Sept. 11, 2008 NAME: (print!) Section: E-MAIL ADDRESS: (print!) 1. Assuming that lim f (x) = 1 , lim g(x) = 2 , lim h(x) = 1 , x2 x2 x2 evaluate the limit x2 lim 4f (x) 2g(x) + 3h(x) 2. For what value of the constant c is the func...
qASept11.pdfPath: Rutgers >> CALC >> 1 Fall, 2008
Description: Solutions to the QUIZ of Sept. 11, 2008 1. Assuming that x2 lim f (x) = 1 , x2 lim g(x) = 2 , x2 lim h(x) = 1 , evaluate the limit x2 lim 4f (x) 2g(x) + 3h(x) Sol. to 1. x2 lim 4f (x) 2g(x) + 3h(x) = lim 4f (x) + lim 2g(x) + lim 3h(x) x...
qSept15.pdfPath: Rutgers >> CALC >> 1 Fall, 2008
Description: QUIZ for Sept. 15, 2008 NAME: (print!) Section: E-MAIL ADDRESS: (print!) 1. Evaluate the limit if it exists: x2 lim 2x + 5 3 x2 2. Evaluate x0 lim sin 10x 5x . ...
qASept15.pdfPath: Rutgers >> CALC >> 1 Fall, 2008
Description: Solutions to the QUIZ for Sept. 15, 2008 1. Evaluate the limit if it exists: x2 lim 2x + 5 3 x2 Solution: First plug-in x = 2 and see what happens. You get 0/0. So this is indeterminate, and we must simplify. The conjugate of 2x + 5 3 is 2x ...
qSept18.pdfPath: Rutgers >> CALC >> 1 Fall, 2008
Description: QUIZ for Sept. 18, 2008 NAME: (print!) Section: E-MAIL ADDRESS: (print!) 1. Exlain why x4 + x 3 has a real root in the open interval 1 < x < 2. 2. Prove rigorously that x1 lim 2x + 1 = 3 . ...
qASept18.pdfPath: Rutgers >> CALC >> 1 Fall, 2008
Description: Solutions to the QUIZ of Sept. 18, 2008 1. Explain why x4 + x 3 has a real root in the open interval 1 < x < 2. Solution: We are going to use the Intermediate Value Theorem (IVT). 1) f (x) is continuous (since it is a polynomial). 2) Plugging-in at ...
qSept22.pdfPath: Rutgers >> CALC >> 1 Fall, 2008
Description: QUIZ for Sept. 22, 2008 NAME: (print!) Section: E-MAIL ADDRESS: (print!) 1. Compute f (2), if f (x) = x2 using the limit denition. , 2. Compute f (x) from the limit denition (no credit for other methods!), where f (x) = 1 x+1 . ...
qASept22.pdfPath: Rutgers >> CALC >> 1 Fall, 2008
Description: Solutions to the QUIZ for Sept. 22, 2008 1. Compute f (2), if f (x) = x2 using the limit denition. Solution to 1: Recall the denition f (a) = lim In this problem a = 2, f (x) = x2 . So f (2) = lim (2 + h)2 22 22 + 2 2 h + h 2 4 4 + 4h + h2 4 = l...
qSept25.pdfPath: Rutgers >> CALC >> 1 Fall, 2008
Description: QUIZ for Sept. 25, 2008 NAME: (print!) Section: E-MAIL ADDRESS: (print!) 1. Dierentiate y= s+2 s2 1 . 2. A particle moves according to a law of motion s = t2 4t + 8, t > 0, where t is measured in seconds and s is measured in feet. (a) Find t...
qASept25.pdfPath: Rutgers >> CALC >> 1 Fall, 2008
Description: Solutions to the QUIZ for Sept. 25, 2008 1. Dierentiate y= Sol. to 1: By the quotient rule s+2 s2 1 . dy (s2 1)(s + 2) (s + 2)(s2 1) (s2 1) 1 (s + 2)(2s) = = ds (s2 1)2 (s2 1)2 Using algebra, this equals s2 1 2s2 4s s2 4s 1 s2 + 4s ...
qSept29.pdfPath: Rutgers >> CALC >> 1 Fall, 2008
Description: QUIZ for Sept. 29, 2008 NAME: (print!) Section: E-MAIL ADDRESS: (print!) 1. Find the rst, second, and third derivatives of the function f (x) = x3 + 2x + 6ex . 2. Dierentiate y= 2 + sin x 1 + 2 cos x . ...
qASept29.pdfPath: Rutgers >> CALC >> 1 Fall, 2008
Description: ...
qOct2.pdfPath: Rutgers >> CALC >> 1 Fall, 2008
Description: QUIZ for Oct. 2, 2008 NAME: (print!) Section: E-MAIL ADDRESS: (print!) 1. Use the rules of dierentiation to calculate f (x) (Do not simplify your answer). e x+1 f (x) = x e + 11 1 2. Find the equation of the tangent line at the point (1, 1) to t...
qAOct2.pdfPath: Rutgers >> CALC >> 1 Fall, 2008
Description: Solutions to the QUIZ for Oct. 2, 2008 1. Use the rules of dierentiation to calculate f (x) (Do not simplify your answer). f (x) = e x+1 x + 11 e 1 Sol. 1: First use the quotient rule: (e x+1 ) (ex + 11) e x+1 (ex + 11) f (x) = (ex + 11)2 Since thi...
qOct6.pdfPath: Rutgers >> CALC >> 1 Fall, 2008
Description: QUIZ for Oct. 6, 2008 NAME: (print!) Section: E-MAIL ADDRESS: (print!) 1. Use rules of dierentiation to calculate f (x) (Do not simplify your answer). f (x) = sin1 x ln x 2. Use logarithmic dierentiation to nd the derivative of the function f (...
qAOct6.pdfPath: Rutgers >> CALC >> 1 Fall, 2008
Description: Solutions to QUIZ for Oct. 6, 2008 1. Use rules of dierentiation to calculate f (x) (Do not simplify your answer). f (x) = sin1 x ln x Solution to 1. First use the product rule: f (x) = (sin1 x)1/2 ln x) = (sin1 x)1/2 ) ln x + (sin1 x)1/2 (ln x)...
qOct16.pdfPath: Rutgers >> CALC >> 1 Fall, 2008
Description: QUIZ for Oct. 16, 2008 NAME: (print!) Section: E-MAIL ADDRESS: (print!) 1. If three resistors with resistence R1 and R2 and R3 are connected in parallel, then the total resistance is given by 1 1 1 1 = + + . R R1 R2 R3 If R1 , R2 , R3 are increas...
qAOct16.pdfPath: Rutgers >> CALC >> 1 Fall, 2008
Description: Solution to theQUIZ for Oct. 16, 2008 Question: If three resistors with resistence R1 and R2 and R3 are connected in parallel, then the total resistance is given by 1 1 1 1 = + + . R R1 R2 R3 If R1 , R2 , R3 are increasing at a rate of 1, 2, and 3 o...
qOct20.pdfPath: Rutgers >> CALC >> 1 Fall, 2008
Description: QUIZ for Oct. 20, 2008 NAME: (print!) Section: E-MAIL ADDRESS: (print!) 1. Let f (x) = to f (4). 4 + x. Using the linear approximation of f (x) at a = 5 compute an approximation 2. Use Newtons method with x1 = 2 to nd the second approximation ...
qAOct20.pdfPath: Rutgers >> CALC >> 1 Fall, 2008
Description: ...
qOct23.pdfPath: Rutgers >> CALC >> 1 Fall, 2008
Description: QUIZ for Oct. 23, 2008 NAME: (print!) Section: E-MAIL ADDRESS: (print!) 1. Find the absolute maximum and the absolute minimum of f (x) = x4 4x + 1 for 0 x 2. 2. Find the critical numbers of the function f (x) = 2x3 9x2 + 12x 2. ...
qAOct23.pdfPath: Rutgers >> CALC >> 1 Fall, 2008
Description: Solutions to the QUIZ for Oct. 23, 2008 1. Find the absolute maximum and the absolute minimum of f (x) = x4 4x + 1 for 0 x 2. Solution of 1: f (x) = 4x3 4. Setting this to 0, we have to solve 4(x3 1) = 0 , i.e. x3 = 1, that gives x = 11/3 = 1....
qOct27.pdfPath: Rutgers >> CALC >> 1 Fall, 2008
Description: QUIZ for Oct. 27, 2008 NAME: (print!) Section: E-MAIL ADDRESS: (print!) 1. Suppose that f (x) is dierentiable everywhere and we know that f (2) = 1 and f (x) 2 for all x. a) What is the largest possible value for f (4) ? b) Show that f (x) has a...
qAOct27.pdfPath: Rutgers >> CALC >> 1 Fall, 2008
Description: Solutions to the QUIZ for Oct. 27, 2008 1. Suppose that f (x) is dierentiable everywhere and we know that f (2) = 1 and f (x) 2 for all x. a) What is the largest possible value for f (4) ? b) Show that f (x) has a root in [2, 4]. Solution to 1a): By...
qOct30.pdfPath: Rutgers >> CALC >> 1 Fall, 2008
Description: QUIZ for Oct. 30, 2008 NAME: (print!) Section: E-MAIL ADDRESS: (print!) 1. Sketch the curve y = 2x3 9x2 + 12x 2 . 2. Sketch the curve y= x2 1 1 . ...
qAOct30.pdfPath: Rutgers >> CALC >> 1 Fall, 2008
Description: Solutions to theQUIZ for Oct. 30, 2008 1. Sketch the curve y = 2x3 9x2 + 12x 2 . Solution: y = 6x2 18x + 12, y = 12x 18. To get the critical points (numbers), we solve y = 0, getting 6x2 18x + 12 = 0, which is the same as 6(x 1)(x 2) = 0 yiel...
qNov3.pdfPath: Rutgers >> CALC >> 1 Fall, 2008
Description: QUIZ for Nov. 3, 2008 NAME: (print!) Section: E-MAIL ADDRESS: (print!) 1. A farmer has 3000 meters of fence, and needs to make a rectangular animal-shed, with a partition in the middle, parallel to one of the sides, using the same fencing materia...
qANov3.pdfPath: Rutgers >> CALC >> 1 Fall, 2008
Description: Solutions to the QUIZ for Nov. 3, 2008 1. A farmer has 3000 meters of fence, and needs to make a rectangular animal-shed, with a partition in the middle, parallel to one of the sides, using the same fencing material. What are the dimensions of the r...
qNov6.pdfPath: Rutgers >> CALC >> 1 Fall, 2008
Description: QUIZ for Nov. 6, 2008 NAME: (print!) Section: E-MAIL ADDRESS: (print!) 1. Evaluate x2 3x + 1 x+ e2x x lim . 2. Evaluate ln x x x lim . ...
qANov6.pdfPath: Rutgers >> CALC >> 1 Fall, 2008
Description: Solutions to theQUIZ for Nov. 6, 2008 1. Evaluate x2 3x + 1 x+ e2x x lim . Solution of 1: We use LHpital twice: o x2 3x + 1 2x 3 2 = lim = lim 2x x 2x 1 x+ x+ 2e x+ 4e2x e lim Now is the time to plug-in, and we get: 2 1 2 = = =0 2x x+ 4e 4e...
qNov10.pdfPath: Rutgers >> CALC >> 1 Fall, 2008
Description: QUIZ for Nov. 10, 2008 NAME: (print!) Section: E-MAIL ADDRESS: (print!) d2 y dx2 dy dx (0) 1. Find y = y(x) if = 6x, = 0 and y(0) = 0. 2. Evaluate the following limit-Riemann sums by any method you like. n n lim i=1 3i n i n 2 1 n . ...
qANov10.pdfPath: Rutgers >> CALC >> 1 Fall, 2008
Description: Solutions to the QUIZ for Nov. 10, 2008 1. Find y = y(x) if d2 y dx2 = 6x, dy dx , dy dx (0) = 0 and y(0) = 0. Solution to 1: To get we take the anti-derivative of 6x: y (x) = 6x dx = 6x2 = 3x2 + C 2 . To nd C, we plug-in x = 0 and get, on the ...
qNov13.pdfPath: Rutgers >> CALC >> 1 Fall, 2008
Description: QUIZ for Nov. 13, 2008 NAME: (print!) Section: E-MAIL ADDRESS: (print!) 1. Find 10 100 x2 dx 10 . 2. Using graphic analysis, nd the denite integral 1 1 x5 dx x4 + 3x2 + 1 (You must justify your answer). ...
qANov13.pdfPath: Rutgers >> CALC >> 1 Fall, 2008
Description: Solutions to the QUIZ for Nov. 13, 2008 1. Find 10 100 x2 dx 10 . Solution to 1: By the area denition of the denite integral, this is the area under the curve y = 100 x2 , above the interval [10, 10]. But this is a semi-circle of radius 100 ...
qANov17.pdfPath: Rutgers >> CALC >> 1 Fall, 2008
Description: Solutions to the QUIZ for Nov. 17, 2008 1. Use the Linear Approximation to approximate 49.1. Solution to 1: We are supposed to use the formula L(x) = f (a) + f (a)(x a) , First, we have to decide on f (x). This is f (x) = 49 = 7 is nice, it is ...
qNov24.pdfPath: Rutgers >> CALC >> 1 Fall, 2008
Description: QUIZ for Nov. 24, 2008 NAME: (print!) Section: E-MAIL ADDRESS: (print!) 1. Compute ecos 2x sin 2x dx 2. Compute /4 sin4 2x cos 2x dx 0 ...
qANov24.pdfPath: Rutgers >> CALC >> 1 Fall, 2008
Description: Solutions to the QUIZ for Nov. 24, 2008 1. 1 4 1 1 3 x+ 3 + x x dx Sol. to 1: First, translate it into power notation: 4 3x1/2 + x3 + x1/2 dx 1 Now, integrate, piece-by-piece, using xn+1 +C . n+1 (except, for this kind of problems, you dont b...
qNov25.pdfPath: Rutgers >> CALC >> 1 Fall, 2008
Description: QUIZ for Nov. 25, 2008 NAME: (print!) Section: E-MAIL ADDRESS: (print!) 1. Compute : 3 0 x2 dx +9 . 2. Compute 2 5x dx 0 . ...
qANov25.pdfPath: Rutgers >> CALC >> 1 Fall, 2008
Description: Solutions to the QUIZ for Nov. 25, 2008 1. Compute ecos 2x sin 2x dx . Sol. to 1: The natural substitution is u = cos 2x (note: not u = 2x). Dierentiating, we get (by the chain rule) du = 2 sin 2x , dx So here is the dictionary u = cos 2x Performing...