2022 CSAT Mathematics (Elective - Calculus)

What is the value of ?

For the function , what is the value of ?

For the arithmetic sequence , Find the value of .

The graph of the function is shown as follows. What is the value of ?

For a sequence with first term 1, where for all natural numbers : Find the value of .

How many integers make the equation have three distinct real roots?

For where , if , what is the value of ?

When the line bisects the area of the region enclosed by the curve and the line , what is the value of the constant ?

The line intersects the graphs of the two functions at points and respectively. When , what is the value of the constant ?

For a cubic function , when the tangent line at point on the curve coincides with the tangent line at point on the curve , what is the value of ?

For a positive number , there is a function defined on the set : As shown in the figure, there is a line passing through three points , , and on the graph of the function . Let be the point other than where the line passing through point and parallel to the -axis intersects the graph of the function . When triangle is an equilateral triangle, what is the area of triangle ? (Note: is the origin.)

A function continuous on the set of all real numbers satisfies for all real numbers . Given that the maximum value of function is 1 and the minimum value is 0, find the value of .

For two constants , the -intercept of the line passing through the two points on the coordinate plane is equal to the -intercept of the line passing through the two points . For the function , when , what is the value of ?

A point moves on a number line, and its position at time is given by two constants as: When the velocity of point at time satisfies , which of the following statements from the options are correct? A. B. There exists in the open interval such that . C. If for all where , then there exists in the open interval such that .

There are two circles with centers respectively and radius length . As shown in the figure, there are three distinct points on circle and a point on circle , where the three points and the three points are each on a straight line. Let . The following is the process of finding the ratio of the lengths of segments and when and . Since , we have and from we get , so . At this time, since , triangles and are congruent. Let Since , we have , and since , we have . In triangle , since , by the cosine law, . Since , . Let the expressions that fit in (A) and (C) above be and respectively, and let the number that fits in (B) be . What is the value of ?

Find the value of .

For the function , given that and , find the value of .

For the sequence , Find the value of .

Find the maximum value of the real number such that the function is increasing on the entire set of real numbers.

A function that is differentiable on the set of all real numbers satisfies the following conditions. (a) On the closed interval , . (b) For some constants , on the interval , . Find the value of .

The sequence satisfies the following conditions. (a) (b) For all natural numbers , (c) Find the value of . [4 points]

Let be a cubic function with leading coefficient and let be the number of real roots of the equation in the closed interval for a real number . The function satisfies the following conditions. (A) For all real numbers , . (B) Find the value of .

What is the value of ?

A function that is differentiable on the set of all real numbers satisfies for all real numbers . What is the value of ?

For a geometric sequence , Find the value of .

What is the value of ?

A point moves on the coordinate plane such that at time , its position is the midpoint of two distinct points where the curve and the line intersect. What is the distance that point travels from time to ?

For the function , let the function be How many values of in the interval make the function have a local minimum?

As shown in the figure, there is a semicircle with diameter of length 2. On arc , two points and are chosen such that and , and let be the intersection point of line segments and . Choose point on line segment , point on line segment , and point on line segment such that line segment is parallel to line segment and triangle is an equilateral triangle. Let be the area of the region enclosed by line segments , , and arc , and let be the area of triangle STU. Then . Find the value of . (where and and are relatively prime natural numbers.)

An increasing and differentiable function on the set of all real numbers satisfies the following conditions. (a) (b) When the inverse function of is , for all real numbers , . When , find the value of . (where and are coprime natural numbers.)