Advanced Placement (AP) · Practice problems

1.1 Change at an instant

30 written problems across three levels — 8 foundational · 9 intermediate · 13 advanced. Work through them on paper, then reveal every answer to check. These are not graded and are counted separately from adaptive practice.

Foundational

1. FoundationalA table of a function $g$ gives $g(1)=1$ and $g(3)=9$ . Estimate the instantaneous rate of change of $g$ at $x=2$ using the symmetric difference quotient.

2. FoundationalUsing the limit of the difference quotient $\lim_{h\to 0}\dfrac{f(-2+h)-f(-2)}{h}$ , find the instantaneous rate of change of $f(x)=x^2 - 2x$ at $x=-2$ .

3. FoundationalOn the curve $y=x^2$ , a secant line joins the points where $x=0$ and $x=1$ . Find the slope of this secant line.

4. FoundationalA table of a function $g$ gives $g(2)=4$ and $g(4)=16$ . Estimate the instantaneous rate of change of $g$ at $x=3$ using the symmetric difference quotient.

5. FoundationalUsing the limit of the difference quotient $\lim_{h\to 0}\dfrac{f(0+h)-f(0)}{h}$ , find the instantaneous rate of change of $f(x)=x^2 - 4x$ at $x=0$ .

6. FoundationalOn the curve $y=x^2$ , a secant line joins the points where $x=1$ and $x=4$ . Find the slope of this secant line.

7. FoundationalA table of a function $g$ gives $g(0)=0$ and $g(2)=4$ . Estimate the instantaneous rate of change of $g$ at $x=1$ using the symmetric difference quotient.

8. FoundationalUsing the limit of the difference quotient $\lim_{h\to 0}\dfrac{f(0+h)-f(0)}{h}$ , find the instantaneous rate of change of $f(x)=x^2 + 3x$ at $x=0$ .

Intermediate

9. IntermediateWater drains from a tank; $V(t)$ is the volume in litres after $t$ minutes. Given $V'(3)=-2$ , which interpretation is correct?

10. IntermediateWhich expression gives the instantaneous rate of change of $f$ at $x=-3$ ?

11. IntermediateDrag $Q$ toward $P=(2,\,4)$ on $y=x^2$ . The secant slope $\dfrac{f(2+h)-f(2)}{h}=2(2)+h$ gives $h=1\Rightarrow 5$ , $h=0.5\Rightarrow 4.5$ , $h=0.1\Rightarrow 4.1$ . Predict the value it approaches as $h\to 0$ : what is the instantaneous rate of change at $P$ ?

12. IntermediateDrag $Q$ toward $P=(4,\,16)$ on $y=x^2$ . The secant slope $\dfrac{f(4+h)-f(4)}{h}=2(4)+h$ gives $h=1\Rightarrow 9$ , $h=0.5\Rightarrow 8.5$ , $h=0.1\Rightarrow 8.1$ . Predict the value it approaches as $h\to 0$ : what is the instantaneous rate of change at $P$ ?

13. IntermediateOn the graph of $f$ , the tangent line at $P=(1,\,-3)$ also passes through $(2,\,-5)$ . What is the instantaneous rate of change of $f$ at $x=1$ ?

14. IntermediateA particle's position is $s(t)=t^2 + 4t$ metres at time $t$ seconds. Find its instantaneous velocity at $t=4$ (in metres per second).

15. IntermediateA particle's position is $s(t)=t^2 + 2t$ metres at time $t$ seconds. Find its instantaneous velocity at $t=1$ (in metres per second).

16. IntermediateA particle's position is $s(t)=t^2 + 5t$ metres at time $t$ seconds. Find its instantaneous velocity at $t=1$ (in metres per second).

17. IntermediateWhich expression gives the instantaneous rate of change of $f$ at $x=3$ ?

Advanced

18. AdvancedFor $f(x)=x^2$ , the instantaneous rate of change equals $10$ at exactly one point. At what $x$ -value does this happen?

19. AdvancedLet $f(x)=a\,x^2$ for an unknown constant $a$ . The instantaneous rate of change of $f$ at $x=-3$ is $18$ . Find $a$ .

20. AdvancedA particle has position $s(t)=t^2$ . Over $[1,\,3]$ its average velocity is $4$ . At what time $t$ in the interval does the INSTANTANEOUS velocity equal this average velocity?

21. AdvancedLet $f(x)=x^2$ and $g(x)=3x^3$ . For $x>0$ , find the value of $x$ at which $f$ and $g$ have the SAME instantaneous rate of change.

22. AdvancedA quadratic $f(x)=x^2+bx+c$ satisfies two conditions: $f(0)=1$ , and its instantaneous rate of change at $x=-1$ is $3$ . Find $f(1)$ .

23. AdvancedTo find the instantaneous rate of $f(x)=x^2$ at $x=2$ , a student computes $\dfrac{f(2+1)-f(2)}{1}=5$ . Which best identifies the error?

24. AdvancedFor $f(x)=x^2$ , the average rate of change over $[a,\,a+4]$ equals the instantaneous rate of change at $x=2$ . Find $a$ .

25. AdvancedTangent lines are drawn to the graph of $f$ at three points: at $P$ the tangent rises -3 units per unit right; at $Q$ the tangent rises 1 unit per unit right; at $R$ the tangent rises -1 unit per unit right. At which point is the instantaneous rate of change GREATEST?

26. AdvancedA quadratic $f(x)=x^2+bx+c$ satisfies two conditions: $f(0)=1$ , and its instantaneous rate of change at $x=-2$ is $-2$ . Find $f(1)$ .

27. AdvancedThe instantaneous rate of change of $f$ at $x=c$ is $-4$ . Which statement MUST be true?

28. AdvancedFor $f(x)=x^2$ , the instantaneous rate of change equals $-2$ at exactly one point. At what $x$ -value does this happen?

29. AdvancedOn an interval, $f$ is increasing and its graph is getting steeper (concave up). Which describes the instantaneous-rate function $f'$ on that interval?

30. AdvancedA particle has position $s(t)=t^2$ . Over $[3,\,7]$ its average velocity is $10$ . At what time $t$ in the interval does the INSTANTANEOUS velocity equal this average velocity?