1. The derivative of the function $y = \sqrt [ 5 ] { x ^ { 4 } }$ is
Answer: C
2. If $f ( x ) = \cos x$, then $f ^ { \prime } \left( \frac { \pi } { 2 } \right)$ has the value ( )
Answer: C
5. It is known that the slope of the tangent line to the graph of the function $f ( x ) = \frac { 2 } { x } - a x$ at the point $( - 1 , f ( - 1 ) )$ is ${ } _ { 1 }$, and the equation of this tangent line is
Answer: A
6. If the function $f ( x ) = \sin x$, then $f \left( \frac { \pi } { 4 } \right) + f ^ { \prime } \left( \frac { \pi } { 4 } \right) = ( )$
Answer: B
7. "$a \leq 0$" is the "Existence of extremes in function $f ( x ) = a x + \ln x$". is neither sufficient nor necessary
Answer: B
8. The following formula is correct
Answer: B
9. The derivative of the function $y = f ( x )$ at $x = x _ { 0 }$ is known to be $f ^ { \prime } \left( x _ { 0 } \right) = - 1$, then $\lim _ { \Delta x \rightarrow 0 } \frac { f \left( x _ { 0 } + 2 \Delta x \right) - f \left( x _ { 0 } \right) } { \Delta x } =$ $\_\_\_\_$.
Answer: D
10. It is known that the derivative of the function $f ( x )$ at $x = 1$ is 1, then $\lim _ { \Delta x \rightarrow 0 } \frac { f ( 1 + \Delta x ) - f ( 1 ) } { 3 \Delta x } =$
Answer: C
11. It is known that the derivative function of $f ( x ) = \frac { 2 x + 1 } { x ^ { 2 } }$ is $f ^ { \prime } ( x )$ and $f ^ { \prime } ( i ) = ( i$ is in imaginary units $)$.
Answer: D
12. $\int _ { 0 } ^ { 3 } \left( x ^ { 2 } + 4 \right) d x =$
Answer: C
13. If the function $y = f ( x )$ is derivable on R and satisfies $x f ^ { \prime } ( x ) + f ( x ) > 0$ with constants ${ } ^ { a }$ , $b$ $\_\_\_\_$ ), then The following inequality must hold
Answer: A
14. The function $f ( x )$ is known to be an even function with domain R. When $x > 0$ is $f ^ { \prime } ( x ) < 0$, then the solution set of inequality $f \left( x ^ { 2 } - x \right) - f ( x ) > 0$ is
Answer: B
The figure is a round table filled with water, if a hole is opened at the bottom, and the volume of water flowing out of any equal time interval is equal, the height of the water in the container $h$ with the time $t$ as a function of $h = f ( t )$, the definition of domain is $D$, and let $t _ { 0 } \in D , t _ { 0 } \pm \Delta t \in D , k _ { 1 } , k _ { 2 }$ denote the average rate of change of $^ { f ( t ) }$ over the interval $^ { \left[ t _ { 0 } - \Delta t , t _ { 0 } \right] , \left[ t _ { 0 } , t _ { 0 } + \Delta t \right] ( \Delta t > 0 ) }$, respectively, and then  size relations
Answer: A
16. The slope of the tangent line to a point $P$ on the curve $y = \frac { 1 } { x }$ is - 4, and the coordinates of point $P$ are
Answer: B
17. The monotonically decreasing interval of the function $y = \frac { 1 } { 2 } x ^ { 2 } - \ln x$ is ( ).
Answer: B
18. The function $f ( x ) = x ^ { 3 } + a x ^ { 2 } + b x + c$, where $a , b , c$ is a real number, and when $a ^ { 2 } - 3 b < 0$ is $f ( x )$ is a function of $a ^ { 2 } - 3 b < 0$ on R
Answer: A
19. If the function $f ( x ) = k x + \mathrm { e } ^ { 2 x }$ is monotonically increasing on the interval ${ } ^ { ( - 1,1 ) }$, then the range of the real number $k$ is ( )
Answer: B
20. If the function $f ( x ) = \ln x + a x ^ { 2 } - 2 x$ is monotonically increasing in the interval $^ { ( 1,2 ) }$, then the real number $a$ is in the range ( )
Answer: D
21. In the sport of high diving, the height of an athlete's center of gravity relative to the water's surface at $t$ (in seconds) $h$ (in: meters) satisfies the relation ${ } ^ { h ( t ) = a t ^ { 2 } + 5 t + 11 }$, and the average rate of change of $h$ when $1 \leq t \leq 2$ is - 10 m/s, then when $t = 3$ when $h$'s instantaneous rate of change is ( )
Answer: C
23. The collinear line of the parabola ${ } ^ { 2 } = 4 y$ intersects the ${ } ^ { y }$ axis at ${ } ^ { M }$, and the equation of the line that is tangent to the parabola at the point ${ } ^ { M }$ is ( ).
Answer: D
24. If $f ( x ) = - \frac { 1 } { 2 } x ^ { 2 } + 2 x f ^ { \prime } ( 2019 ) - 2019 \ln x$ is known, $f ^ { \prime } ( 1 ) =$ is known.
Answer: D
25. The function $f ( x ) = \frac { 1 } { 3 } x ^ { 3 } + x + \cos x$ is known, and if $f ( 1 ) > f \left( \log _ { 2 } x \right)$, then the range of values of the real number ${ } _ { x }$ is ( ).
Answer: A
26. It is known that the function $f ( x ) = e ^ { x } + f ^ { \prime } ( 0 ) x ^ { 2 } + 3 x + 2$, then $f ^ { \prime } ( 1 ) = ( )$
Answer: C
27. The graph of the function $f ( x ) = a e ^ { x } + b x - 1$ is known to be tangent to the $x$ axis at the origin of the coordinates, then the values of $a$ and $b$ are ( ) respectively.
Answer: B
28. If $x = \ln 3$ is known to be a minima of the function $f ( x ) = \mathrm { e } ^ { x } + a x$, then $a =$ ( )
Answer: D
29. The function $f ( x ) = \frac { 1 } { x }$ is known to be $\lim _ { \Delta x \rightarrow 0 } \frac { f ( 1 + \Delta x ) - f ( 1 ) } { \Delta x } = ( )$.
Answer: A
31. The following derivation operations are correct
Answer: B
32. A function $y = f ( x )$ is said to have the property $T$ if there exist two points on the image of the function $y = f ( x )$ such that the tangent lines to the image of the function at these points are perpendicular to each other. The following functions have the property $T$.
Answer: A
33. Given that the function $f ( x ) = \ln x - \frac { 1 } { 3 } a x ^ { 3 }$ is monotonically decreasing on $( 1 , + \infty )$, the range of the real number $a$ is
Answer: B
34. The area of the plane formed by the line $x = - 2 , x = 2 , y = 0$ and the curve $y = x ^ { 2 } - x$ is ( )
Answer: B
35. The function $f ( x ) = \sin 2 x - k \sin x + x$ is monotonically decreasing on the interval $\left( 0 , \frac { \pi } { 2 } \right)$, then the value of $k$ is in the range ( ).
Answer: B
36. If the function $f ( x ) = x ^ { 2 } + 2 \cos x$ is known, the solution set of the inequality $f ( 2 x - 1 ) < f ( 3 x )$ is ( ).
Answer: C
37. It is known that the parabola $C : x ^ { 2 } = 4 y$, the line through the point $M ( 0,4 )$ and $C$ intersects at the points $A , B$, and the tangent line at the points $C _ { \text {在 } } A , B$ intersects at the point $N$. Of the following four points, the one that can be the midpoint of the line segment $M N$ is ( )
Answer: B
38. The function $f ( x ) = 2 ( x - 1 ) \mathrm { e } ^ { x } - x ^ { 2 } - a x$ is known to be monotonically increasing on R. The maximum value of $a$ is ( ).
Answer: A
39. If the line $y = a x + b$ is known to be tangent to the curve $\mathrm { y } = \mathrm { x } + \frac { 1 } { \mathrm { x } }$, the maximum value of $2 a + b$ is ( )
Answer: C
40. It is known that the function $f ( x ) = \sin \left( \omega x + \frac { \pi } { 6 } \right) ( \omega > 0 )$ is monotonically decreasing on $( 1,2 )$ and monotonically increasing on $( 2,3 )$, and the circle $x ^ { 2 } + y ^ { 2 } = r ^ { 2 } ( r > 0 )$ contains exactly three points corresponding to the extremes of $f ( x )$. $x ^ { 2 } + y ^ { 2 } = r ^ { 2 } ( r > 0 )$ contains exactly three points corresponding to the extremes of $f ( x )$, then the range of $r$ is ( ) High School Mathematics Assignment, October 29, 2025
Answer: B
导数与微积分
36 Practice Questions
Practice with Chinese questions to prepare for the CSCA exam. You can toggle translations while practicing.
Derivatives and calculus are core tools of mathematical analysis, focusing on the rate of change of functions, the slope of tangents, and area calculations. In the CSCA exam, this part of the content usually appears in the form of calculation and application questions, requiring candidates to master the basic rules of derivatives, the geometric significance of derivatives and simple integral operations. The questions are of moderate difficulty and focus on the application of basic formulas and computational accuracy.
It is recommended that the derivation formula be reinforced through categorical exercises, with special attention to the chain rule for the derivation of composite functions and the standard solution of the tangent line equation.
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