2023 CSAT Mathematics (Elective: Calculus)

What is the value of ?

For a geometric sequence with a positive common ratio that satisfies find the value of .

For a polynomial function , let function be defined as Given that and , what is the value of ?

Given that and , what is the value of ?

The function has a local maximum at and a local minimum at . Find the value of . (where are constants.)

For an arithmetic sequence where all terms are positive and the first term equals the common difference, if is satisfied, what is the value of ?

What is the -intercept of the tangent line drawn from the point to the curve ?

The function has a maximum value of 7 and a minimum value of 3 on the closed interval . Find the value of . (where are constants.)

Let be the area of the region enclosed by the two curves and the -axis, and let be the area of the region enclosed by the two curves and the line . When , what is the value of the constant ? (Given that )

As shown in the figure, quadrilateral is inscribed in a circle and What is the length of the radius of this circle?

A function continuous on the set of all real numbers satisfies the following condition. When , . (where is a natural number) The function defined on the open interval has a minimum value of 0 at . Find the value of .

For a natural number , let be the number of natural numbers such that there exists an integer among the -th roots of . Find the value of .

For a polynomial function , function is defined as follows: For the function , which of the following statements in are correct? A. B. Function is continuous on the set of all real numbers. C. If function is decreasing on the closed interval and , then function has a minimum value on the set of all real numbers.

For all sequences where all terms are natural numbers and satisfy the following conditions, let and be the maximum and minimum values of , respectively. Find the value of . (a) (b) For all natural numbers :

Find the value of real number that satisfies the equation

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

For two sequences , Find the value of .

Find the number of integers such that the equation has exactly 2 distinct positive real roots.

A point P moving on a number line has velocity and acceleration at time that satisfy the following conditions. (a) When , . (b) When , . Find the distance traveled by point P from time to .

For a natural number , let the function be defined as For a real number , let be the number of distinct real roots of the equation in . Find the sum of all natural numbers such that the maximum value of the function is 4.

Given that is a cubic function with leading coefficient 1 and is a function continuous on the set of all real numbers, find the value of when the following conditions are satisfied: (a) For all real numbers : (b) The minimum value of function is . (c)

What is the value of ?

What is the value of ?

For a geometric sequence , if , what is the value of ?

For three constants , the function satisfies the following conditions. (condition 1) (condition 2) When the inverse function of is denoted as , we have . Find the value of . (where are rational numbers and is irrational.)

For a cubic function with a positive leading coefficient and the function , the composite function defined on the set of all real numbers satisfies the following conditions. (a) The function has a local maximum value of 0 at . (b) In the open interval , the number of distinct real roots of the equation is 7. Given that , we have . Find the value of . (Here, and are relatively prime natural numbers.)