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Advanced Placement (AP) · Practice problems

1.1 Change at an instant

10 problems, including 4 🔥 challenging (exam-level, multi-step). Work through them on paper, then reveal every answer to check.

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

  4. 4. Which expression gives the instantaneous rate of change of f at x=-3?

  5. 5. Drag 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?

  6. 6. Drag 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?

    7. 🔥 Challenging problems

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

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

  9. 9. 🔥 ChallengingA 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?

  10. 10. 🔥 ChallengingLet 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.