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Tasks include implicit differentiation, analyzing a graph that represents f'(x) to determine intervals of increase and concavity, parameterizing a line segment, sketching piecewise-defined functions, and forming tangent line equations from parametric curves. Additional sections cover limits, critical points and monotonicity for f(x)=x e^{-2x^2}, and maximizing rectangle area under a curve, supported by a collection of differentiation and calculus formulas.",{"@graph":69,"@context":122},[70,84,105],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":20,"@type":76,"position":81},"https://docshare.wps.com/document/exam/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/pre-calculus-review-set-problems-80-and-124-pre-calculus-review/453895/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/pre-calculus-review-set-problems-80-and-124-pre-calculus-review/453895.png","ImageObject",300,407,{"name":92,"@type":93},"SundayMiles","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-20","2026-09-30",true,{"@type":102,"interactionType":103,"userInteractionCount":44},"InteractionCounter",{"@type":104},"ViewAction",{"@type":106,"mainEntity":107},"FAQPage",[108,114,118],{"name":109,"@type":110,"acceptedAnswer":111},"What is required for the derivative problems in this set?","Question",{"text":112,"@type":113},"Derivatives must be found with all necessary work shown. Answers do not have to be fully simplified when solving for derivatives.","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"How do you analyze the graph of f'(x) to answer questions about f(x)?",{"text":117,"@type":113},"Use the sign of f'(x) to identify where f(x) is increasing, and use concavity information derived from f'(x) behavior to find intervals where f(x) is concave up.",{"name":119,"@type":110,"acceptedAnswer":120},"What optimization problem is included and what must be determined?",{"text":121,"@type":113},"A rectangle is placed under the curve y=√x with its sides parallel to the axes, left at x=a and right at x=b where b is fixed at 20. Determine the value of a that gives the maximum possible area and the maximum area as exact values.","https://schema.org",{"og:url":83,"og:type":124,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":126,"canonical":83},"index,follow",{"doc_id":128,"site_id":62},453895,1791092394,{"code":4,"msg":5,"data":131},{"doc_id":128,"user_id":132,"nickname":92,"user_avatar":133,"doc_module":4,"category_id":19,"category_name":20,"doc_title":65,"doc_description":67,"doc_content":134,"file_id":135,"file_url":136,"file_type":137,"file_size":138,"view_count":44,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":44,"language":139,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":140,"faqs":141,"seo_title":142,"seo_description":67,"update_tm":143,"read_time":144},1099529839202,"https://ap-avatar.wpscdn.com/davatar_a8503ba1806abce46bf441b54a3ca4cd","(1) (Pre-calculus Review Set Problems 80 and 124.)  \n(a) Determine if each of the following statements is True or False. If it is true, explain why. If it is false, give a counterexample.  \n(i) If a and b are real numbers and 2a5 b7 = 3ab, then 2a4 b6 = 3 .   .  \n(ii) When solving x2 (x − 2)3 = (x − 2)3 , we get x2 = 1, so the solutions are x 1 = 1 , x2 = −1 .   .  \n(b) Simplify and write the following expression without negative exponents. Show your work.  \n6−1r −3r 2  \nr 5  \n(2) Find the derivatives of the given functions. You must show your work. But you do not have to simplify your answers.  \n(a) z (t) = pt (t2 + t +5)  \n(b) h(x) = ✓5x2 − pxx◆7  \n(c) (6 points) Find dy/dx implicitly: ex2y = x + y  \n(3) (a) (9 points) The graph below is the derivative of a function f(x), f 0 (x) . Answer each of the following questions. You do not need to explain.  \ny  \n4  \nf’(x)  \n −4      4 x  \n−4  \n(i) f (x) is increasing on the interval(s)    \n(If you are not sure about the coordinates of the end points, an estimate will do.)  \n(ii) f (x) is concave up on the interval(s)   .  \n(iii) The function f(x) achieves its absolute maximum value on the interval [−4 , 4] at x =   .  \n(b) (6 points) Write a parameterization for the line segment from (2 , −5) and (−1 , 3) .  \n8 |x − 1| , 0 􀀣 x 􀀣 2  \n(c) (6 points) Consider the function: f(x) = \u003C .  \n: 3 − x, 2 \u003C x 􀀣 4  \n(i) Sketch the graph of f (x) . Please label clearly the value of f at x = 0 , 1 , 2 , 3 , and 4 .  \n(ii) Find the average value of f(x) on the interval [0 , 4] .  \n(d) (6 points) Find the equation of the tangent line (in the form of y = mx + b) at the point (2 , 1) to the curve defined by the parametric equations  \nx = 2t, y = t3 .  \n(4) Find each of the following limits when it exists, write DNE otherwise. Show your work.  \n13 + 2x − 4x4 +3x4  \n(a) lim  \nx!1 2x − 5x2 − 5x4  \n x − a (b) lim   , where a > 0.  \nx!a+ px2 − a2  \n(ln(x))2  \n(c) lim  \nx!1 x2  \n(5) Consider f(x) = xe−2x2 . Answer each of the following questions. You must show all work.  \n(a) Find f 0 (x) and all critical point(s) of f(x) .  \n(b) Determine the interval(s) where f (x) is increasing and decreasing. State the xcoordinate(s) of the point(s) where f achieves it local maximum or/and local minimum.  \n(c) Find f 00 (x) and all inflection point(s) of f(x) .  \n(6) The figure below shows the curve y = px, and a rectangle with its upper-left corner on the curve, its sides parallel to the axes, its left end at x = a, and its right end at x = b. Let b be fixed as b = 20 . Find the value of a such that the rectangle has the maximum possible area. What is that maximum possible area? Show your work and give exact values.  \nUseful formulas  \n• The derivative of a function  \nf 0 (x) = lim f (x + h) − f(x)  \nh!0 h  \n• Some rules of di↵erentiation  \n d   \ndx (cf(x)) = cf0 (x) ddx (f(x)g (x)) = f0 (x)g (x)+ f(x)g0 (x) ddx ✓ ~~ ~~fg((xx)) ◆ = f~~ ~~0~~ ~~(x)g~~ ~~(xg)~~ ~~(−x)f2(x)g0~~ ~~(x)  \nddxf (g (x)) = f0 (g (x))g0 (x)  \n• Di↵erentiation formulas  \n\n| ddx (xn ) = nxn−1\u003Cbr>ddx (ln x) = 1x\u003Cbr>ddx (arcsin(x)) = p~~ ~~1~~ ~~~~ ~~x2 ddx (sinh(x)) = cosh(x) | ddx (ex ) = exddx (sin(x)) = cos xddx (tan(x)) = sec2 xddx (sec(x)) = sec x tan x  d  (arccos(x)) =  −1  dx p 1 − x2\u003Cbr> d \u003Cbr>dx (cosh(x)) = sinh(x) | ddx (ax ) = (lna)ax ddx (cos(x)) = − sin xddx (cot(x)) = − csc2 xddx (csc(x)) = − cscx cotxddx (arctan(x)) = 1~~ ~~x2 ddx (tanh(x)) = cos1h2~~ ~~(x) |\n| --- | --- | --- |\n\n• The linear approximation of a function f at a is given by  \nf (x) ⇡ f(a)+ f0 (a)(x − a)  \n• Derivative of the inverse function If f is a one-to-one di↵erentiable function withinverse function f −1 and f 0 (f−1(a))  0, then the inverse function of f is di↵erentiable at a and  \nddx &f−1(a)' = f~~ ~~0~~ ~~(f1~~ ~~(a)) .  \n• Parametric Equations for a straight line: An object moving along a line through the point (x0 , y0 ), with dx/dt = a and dy/dt = b has parametric equations  \nx = x0 + at, y = y0 + bt.  \nThe slope of the line is m = b/a.  \n• The instantaneous ","cbCaij5EookOG7le","https://ap.wps.com/l/cbCaij5EookOG7le","pdf",240987,"English","# True or False and Simplification\n# Derivatives and Related Problems\n# Graph of f'(x)\n# Parameterization and Piecewise Functions\n# Tangent Line from Parametric Equations\n# Limits\n# Critical Points, Monotonicity, and Inflection Points\n# Optimization: Maximum Rectangle Area\n# Useful Formulas","[{\"question\":\"What is required for the derivative problems in this set?\",\"answer\":\"Derivatives must be found with all necessary work shown. Answers do not have to be fully simplified when solving for derivatives.\"},{\"question\":\"How do you analyze the graph of f'(x) to answer questions about f(x)?\",\"answer\":\"Use the sign of f'(x) to identify where f(x) is increasing, and use concavity information derived from f'(x) behavior to find intervals where f(x) is concave up.\"},{\"question\":\"What optimization problem is included and what must be determined?\",\"answer\":\"A rectangle is placed under the curve y=√x with its sides parallel to the axes, left at x=a and right at x=b where b is fixed at 20. Determine the value of a that gives the maximum possible area and the maximum area as exact values.\"}]","Pre-Calculus Review Set Problems 80 and 124 - Pre-Calculus Review | PDF",1790739781,23]