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Calculus Exam - University of Houston Math ContestJanuary 30, 20211) If we are told thata)b)then the value ofc)d)ise) None of the other answers provided.2) Supposeis a differentiable, invertible function whose graph passes through the pointwhere itsnormal line has a slope of. What is the slope of the tangent line to the graph of the inverse functionat the point?a)b)c)d)e) None of the other answers provided.3) The graph of the positive functiondetermines a region over the intervalarea (such as the one shown in the Figure 1 below).When this region is revolved about the x-axis the resulting solid enclosesvolume is enclosed by the solid of revolution obtained by revolvinginterval?a)d)units of volumeunits of volumeb)units of volumee) None of the other answers provided.that encloses units ofunits of volume. How muchabout the x-axis along thec)units of volume4) Two particles are moving along the y-axis with positions given, respectively, byandAt how many distinct points in time do the two particles share the same acceleration?a) The accelerations match at three points in time.b) The accelerations match at one point in time.c) The accelerations match at four points in time.d) The accelerations match at two points in time.e) None of the other answers provided.
5) For which value ofa)b)does the following limit hold?c)6) At each pointd)that makes this equation true.along a given differentiable curve there is a tangent line with slopecurve passes through the pointa)e) There is no value ofb), what is the y-coordinate of the curve whenc)7) The differentiable functiond).The lineis tangent to the graph ofwhich of the following statements, if any, are true?a) I and II only.b) I, II and III.e) None of the other answers provided.?e) None of the other answers provided.is given byI. The (signed) area between the graph ofII.III. The average value of over the interval. If theat the point. Based on this information,and the t-axis along the intervalequals .equals .c) II and III only.d) I and III only.8) The value of the limitdepends on the real numberThe smallest value of this limit occurs whena)b)c)d).e) None of the other answers provided.Figure 2 is used for Problems 9-11 and is shown below.9) Given the graph ofa)b)University of Houston Math Contest 2021https://mathcontest.uh.eduin Figure 2, it follows thatc)d)e) None of the other answers provided.Calculus ExamPage 2 of 9
Figure 2 is shown again for use in Problems 10 and 11.10) Letdenote the number of points in the intervalthe number of points in the intervalwhereis shown in figure 2.) Thena)b)c)11) Given the graph ofd)e) None of the other answers provided.in Figure 2 and settinga) Nothing since the proposed limitdoes not exist.e) None of the other answers provided.12) Letwherefails to be differentiable, and letdenote. (Here, as in the previous problem, the functionbe defined by the formulato determine the value of, it follows thatb)c)and letb)d)e) None of the other answers provided.andc)are two positive numbers satisfying the two equationsWe can conclude that the value ofisa)d)b). Use the fact that.a)13) Given thatd)c)University of Houston Math Contest 2021https://mathcontest.uh.edue) None of the other answers provided.Calculus ExamPage 3 of 9
14) Suppose we are told thatexists. Which, if any, of the following statements can be true?I.II.III.a) II and III onlyb) I and III only15) If the functionc) I, II, and IIId) I and II onlye) None of the other answers provided.is continuous for all real numbers is given by the formulafor, thena)b)c)d)e) Undefined16) Several curves are shown below (some that look oval-shaped, others that look like indented eggs, and still othersthat cross themselves or are even disconnected).Each of these curves is an example of a Cassini Oval*, and the one shown in black is given by the equation.The tangent line depicted in the image above passes through the point. The slope of this line equals[*Many Calculus students are familiar with standard ellipses given by equations such as, butare perhaps less familiar with the fact that these ellipses are the set of all points whose sum of distances to two fixedpoints (called the "foci") is constant. A Cassini Oval, on the other hand, is defined as the set of all points whoseproduct of such distances is constant. In the figure above, the two foci are located at.]a)b)University of Houston Math Contest 2021https://mathcontest.uh.educ)d)e) None of the other answers provided.Calculus ExamPage 4 of 9
17) Supposeis a continuous function whose domain and range both equal the interval. One canconclude that the graph ofmust intersect the graph ofat one or more points by applying theIntermediate Value Theorem to which of the following functions?a)b)c)d)e) None of the other answers provided.18) We are given that the following limit holds for some function:.Of the following options provided below, which, if any, could equal the functiona)b)e) None of the other answers provided.19) Given thata)c), the value ofb)c)20) If the piecewise functiond)isd)is defined for all real numbers?e) None of the other answers provided.bythen which, if any, of the following statements are true?a)is continuous everywhere,is differentiable everywhere, andb)is discontinuous andis a local minimum.c)is continuous everywhere,is not differentiable everywhere, andd)is continuous everywhere,is not differentiable everywhere, ande) None of the other answers provided.is a local maximum.is a local maximum.is a local minimum.21) The graph of the differentiable functionhas a tangent line when, and this line passes throughthe pointsand. Based on this information determine which of the following statements is true.a)d)andandb)andc)e) None of the other answers provided.and22) The functionhas a continuous second derivative, and its graph has a tangent line at the pointby. If we are also told that, then amongst the options provided below, which value ofus to conclude thatis a local minimum?a)b)e) There are no values ofc)for whichUniversity of Houston Math Contest 2021https://mathcontest.uh.eduis a local minimum sincegivenallowsd)is not a critical number for.Calculus ExamPage 5 of 9
23) The definite integralsandeach depend on the value of the positive, whole numberwhole number that is divisible by ?. For which values of suchis the productaa) This is true foronly.b) This is not true for any values of .c) This is true for all odd values of .d) This is true for all values of .e) None of the other answers provided.24) Figure 3 shows the graph of a functiontold thatthen what is the value of the numbera)b)that passes through the pointsand. If we are?c)d)e) None of the other answers provided.25) Find an exact solution to the Ordinary Differential Equationa)b)e) None of the other answers provided.with initial conditionc)26) Consider an ellipse given by the equation.d)where. What is the maximum product ofdistances from the focii to a point on the curve?a)b)University of Houston Math Contest 2021https://mathcontest.uh.educ)d)e) None of the other answers provided.Calculus ExamPage 6 of 9
27) The graph of, the derivative ofIfanda)b), is the line shown in Figure 4 below., then what is the value of ?c)d)e) None of the other answers provided.28) The radius of a sphere is increasing at a rate proportional to the value of the radius. If the radius initiallymeasures cm and the radius equals cm two seconds later, how large will the radius be after seconds?a)cm29) Supposeb)cmc)cmd)cme) None of the other answers provided.is everywhere continuous with known limitsWhich, if any, of the following conclusions hold?a)b)University of Houston Math Contest 2021https://mathcontest.uh.educ)d)e) No conclusions may be drawn.Calculus ExamPage 7 of 9
30) Which functionssatisfy the following property: the average value offor every possibleover the intervalequals?I. Any linear function whose graph passes through the origin.II. Any quadratic function whose graph is a parabola with its vertex at the origin.III. Any function that satisfies the separable ODEa) I, II and IIIb) I and II onlye) None of the other answers provided.c) II and III only31) Bizarre as it may seem, the functionDifferential Equation (with initial conditiona)b)e) None of the other answers provided.). Which equation doesc)32) Figure 5 shows the graph of , the derivative of a functionany, of the following statements are true.a) is concave up onand concave down onb) is concave down onand concave up onc)andare local minima for .d) is increasing on the interval.e) None of the other answers provided.University of Houston Math Contest 2021https://mathcontest.uh.edud) I onlysatisfies a lovely Ordinarysatisfy?d). Based on this graph, determine which one, if.Calculus ExamPage 8 of 9
33) Several integral expressions are written below. Which ones correspond to the area of the region shown in Figure6?I.II.III.IV.a) I, III and IV onlyb) I and IV onlyc) II and III onlyd) II and IV onlye) I and III only34) The top of afoot ladder is sliding down a wall at a constant rate of feet per minute, but this information isirrelevant to this question. What is relevant is the fact that atop this falling ladder stood a mathematician who paintedon her wall an antiderivative for the function. Which beautiful function did she paint?a)35) Letb)c)d)e)denote the number of questions from this test that you answered correctly. The functionhas a point of inflection whena)b)e) The function has no points of inflection.University of Houston Math Contest 2021https://mathcontest.uh.educ)equals which number?d)Calculus ExamPage 9 of 9
Calculus Exam - University of Houston Math Contest January 30, 2021 1) If we are told that then the value of is . 2021, Calculus, Math Contest, Mathematics Contest, High School, Middle School, Juni
