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Includes multiple-choice problems on finding minima/maxima using derivatives, solving for coordinates of inflection points, determining intervals where functions are decreasing or concave down, and related word problems such as a ladder sliding down a wall. Part II provides free-response solutions for extreme values using critical points and endpoint comparisons.",{"@graph":69,"@context":123},[70,84,106],{"@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/ap-calculus-chapter-4-testbank-part-i-multiple-choice-questions/459823/",{"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/ap-calculus-chapter-4-testbank-part-i-multiple-choice-questions/459823.png","ImageObject",300,407,{"name":92,"@type":93},"MrHarris58","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-26","2026-09-30",true,{"@type":102,"interactionType":103,"userInteractionCount":105},"InteractionCounter",{"@type":104},"ViewAction",13,{"@type":107,"mainEntity":108},"FAQPage",[109,115,119],{"name":110,"@type":111,"acceptedAnswer":112},"How do you determine local extrema for a polynomial like f(x)=x^3+3x^2-45x+4?","Question",{"text":113,"@type":114},"Compute f'(x), solve for critical points, then classify each point as a local minimum or maximum based on the sign changes in f'(x) (or the values compared to neighboring intervals).","Answer",{"name":116,"@type":111,"acceptedAnswer":117},"How is the volume change of a sphere approximated when the radius increases slightly?",{"text":118,"@type":114},"Use differentials: express volume V in terms of radius r, compute dV=V'(r)dr, and substitute r=10 with dr=10.02-10 to estimate the change.",{"name":120,"@type":111,"acceptedAnswer":121},"What is the role of the Mean Value Theorem in these problems?",{"text":122,"@type":114},"It guarantees the existence of a point c in the interval where f'(c) equals the average rate of change. The tasks then solve for c using that equation on the specified interval.","https://schema.org",{"og:url":83,"og:type":125,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":127,"canonical":83},"index,follow",{"doc_id":129,"site_id":62},459823,1790884670,{"code":4,"msg":5,"data":132},{"doc_id":129,"user_id":133,"nickname":92,"user_avatar":134,"doc_module":4,"category_id":19,"category_name":20,"doc_title":65,"doc_description":67,"doc_content":135,"file_id":136,"file_url":137,"file_type":138,"file_size":139,"view_count":105,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":140,"language":141,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":142,"faqs":143,"seo_title":144,"seo_description":67,"update_tm":145,"read_time":146},687212988360,"https://ap-avatar.wpscdn.com/davatar_276721f389ce27ea32af1340a28f341c","AP Calculus Chapter 4 Testbank  \n(Mr. Surowski)  \nPart I. Multiple-Choice Questions  \n1. Let f(x) = x3 + 3x2 􀀀 45x + 4 . Then the local extrema of f are  \n(A) a local minimum of 􀀀179 at x = 5 and a local maximum of  \n77 at x = 􀀀3 .  \n(B)  \na local minimum of 􀀀77 at x = 3 and a local maximum of 179 at x = 􀀀5  \n.  \n(C) a local minimum of 􀀀179 at x = 􀀀5 and a local maximum of 􀀀77 at x = 3 .  \n(D) a local minimum of 􀀀77 at x = 3 and a local maximum of  \n77 at x = 5 .  \n(E) a local minimum of 77 at x = 􀀀5 and a local maximum of  \n179 at x = 3 .  \n2. Consider the function f(x) = x3 + 3x2 􀀀 45x + 4 . Then  \nI. f is decreasing on (􀀀5 ; 3) and increasing on ( 􀀀1 ; 􀀀5)[(3 ; 1 ) .  \nII. f has a local minimum at x = 3 .  \nIII. f has a local maximum at x = 􀀀5 .  \n(A) I only (B) II only (C) I and II only (D) I and III only  \n(E) I, II, and III  \n3. For what value of x does the function f(x) = x3 􀀀 9x2 􀀀 120x + 6 have a local minimum?  \n(A) 10 (B) 4 (C) 3 (D) 􀀀4 (E) 􀀀10  \n4. Use differentials to approximate the change in the volume of a sphere when the radius is increased from 10 to 10.02 cm.  \n(A) 4213.973 (B) 1261.669 (C) 1256.637 (D) 25.233 (E) 25.133 5. The graph of y = x3 􀀀 5x2 + 4x + 2 has a local minimum at  \n(A) (0:46 ; 2:87)  \n(B) (0:46 ; 0)  \n(C)  \n(2:94 ; 􀀀4:05)  \n(D) (4:06 ; 2:87)  \n(E) (1:66 ; 􀀀0:59)  \n6. Find a positive value, c, for x, that satisfies the conclusion of the Mean Value Theorem for Derivatives for f(x) = 3x2 􀀀 5x + 1 on the interval [2 ; 5] .  \n(A) 1  \n13  \n(B) ~~ ~~  \n6  \n11  \n(C) ~~ ~~  \n6  \n23  \n(D) ~~ ~~  \n6  \n(E)  \n\n| 7 |\n| --- |\n| 2 |\n\n7. The graph of y = x3 􀀀 2x2 􀀀 5x + 2 has a local maximum at (A) (2:120 ; 0)  \n(B) (2:120 ; 􀀀8:061)  \n(C) (􀀀0:786 ; 0)  \n(D)  \n(􀀀0:786 ; 4:209)  \n(E) (0:666 ; 􀀀1:926)  \n8. A 20-foot ladder slides down a wall at 5 ft/sec. At what speed is the bottom sliding out (in ft/sec) when the top is 10 feet from the floor?  \n(A) 0:346 (B) 2:887 (C) 0:224 (D) 5:774 (E) 4:472  \n9. What are the coordinates of the point of inflection on the graph of y = x3 􀀀 15x2 + 33x + 100?  \n(A) (9 ; 0) (B) (5 ; 􀀀48) (C) (1 ; 119) (D) (9 ; 􀀀89) (E)  (5 ; 15)   \n10. The graph of y = x4 + 8x3 􀀀 72x2 + 4 is concave down for  \n(A) 􀀀6 \u003C x \u003C 2  \n(B) x > 2  \n(C) x \u003C 􀀀6  \n(D) x \u003C 􀀀3 􀀀 3p5 or x > 􀀀3 + 3p5  \n(E) 􀀀3 􀀀 3p5 \u003C x \u003C 􀀀3 + 3p5  \n11. The function f is given by f (x) = x4 + 4x3. On which of the following intervals is f decreasing?  \n(A) (􀀀3 ; 0) (B) (0 ; 1 ) (C)  (􀀀3 ; 1 )  (D) ( 􀀀1 ; 􀀀3) (E) ( 􀀀1 ; 0)  \n12. The value of c that satisfies the Mean Value Theorem on the interval [0 ; 5] for the function f(x) = x3 􀀀 6x is  \n5  \n(A) 􀀀 ~~  ~~  p 3  \n(B) 0  \n(C) 1  \n5  \n(D) ~~ ~~  \n3  \n(E)  \n5  \np3  \n13. The graph of the function y = x3 + 12x2 + 15x + 3 has a relative maximum at x =  \n(A) 􀀀10:613 (B) 􀀀:248 (C) 􀀀7:317 (D) 􀀀1:138 (E) 􀀀:683  \n14. The side of a square is increasing at a constant rate of 0.4 cm/sec. In terms of the perimeter, P, what is the rate of change of the area of the square, in cm2 /sec?  \n(A) 0:05P (B) 0:2P (C) 0:4P (D) 6:4P (E) 51:2P  \n15. The second derivative of a function is given by f00 (x) = xsin x 􀀀  \n2. How many points of inflection does f have on the interval (􀀀10 ; 10)?  \n(A) Zero (B) Two (C) Four (D) Six (E) Eight  \nPart II. Free-Response Questions  \n16. Find the extreme values of the function f(x) = x3 +3x2 􀀀 36x +4 on the interval [0 ; 4] .  \nThe critical x-values are obtained by solving f 0 (x) = 0 . This leads to 3x2 + 6x 􀀀 36 = 0 ) x = 􀀀1 +p 13 . The extreme values are obtained by comparing:  \nf (0) = 4  \nf (􀀀1 +p13 ) = 􀀀51:74  \nf (4) = 􀀀28  \nTherefore the maximum of f is 4 and the minimum is 􀀀51:74 .  \n17. Find the extreme values of the function f (x) = x~~2 ~~ on the interval [􀀀3 ; 3] .  \nFrom f 0 (x) = 11~~ ~~~~ ~~xx2~~2~~, we see that the critical x-values on the given interval are x = 􀀆1 . We compare:  \nf (􀀀3) = 30  \n1 f (􀀀1) = 􀀀  \n2 f (1) = 12  \n 3   \nf (3) =  \n10  \nTherefore f has a maximum of 12 (at x = 1) and a minimum of 􀀀 12 (at x = 􀀀1) .  \n18. Find the extreme values of the","cbCaifJM8aKc3NBp","https://ap.wps.com/l/cbCaifJM8aKc3NBp","pdf",203028,23,"English","# Part I. Multiple-Choice Questions\n## Questions 1-15\n# Part II. Free-Response Questions\n## Questions 16-21","[{\"question\":\"How do you determine local extrema for a polynomial like f(x)=x^3+3x^2-45x+4?\",\"answer\":\"Compute f'(x), solve for critical points, then classify each point as a local minimum or maximum based on the sign changes in f'(x) (or the values compared to neighboring intervals).\"},{\"question\":\"How is the volume change of a sphere approximated when the radius increases slightly?\",\"answer\":\"Use differentials: express volume V in terms of radius r, compute dV=V'(r)dr, and substitute r=10 with dr=10.02-10 to estimate the change.\"},{\"question\":\"What is the role of the Mean Value Theorem in these problems?\",\"answer\":\"It guarantees the existence of a point c in the interval where f'(c) equals the average rate of change. The tasks then solve for c using that equation on the specified interval.\"}]","AP Calculus Chapter 4 Testbank - Part I - Multiple-Choice Questions | PDF",1790757012,58]