AP Unit 2 Problems
Thirty problems for AP Unit 2. Try the problem before opening its separate solution. Write the mathematical setup and the reason your method applies.
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Problem 1
For s(t)=t²−2t meters, find the average rate on [1,4] seconds and the instantaneous rate at t=1 using a limiting difference quotient.
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Problem 2
Find the derivative of f(x)=x²+1 at x=−2 using lim(x→−2)[f(x)−f(−2)]/(x+2).
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Problem 3
A bacteria population B(t), measured in thousands, has B(3)=7 and B(3+h)=7+2h+0.4h² near h=0. Find B′(3), with units.
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Problem 4
Use the derivative definition to find f′(x) for f(x)=x²−3x.
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Problem 5
For f(x)=x²+2, find the equation of the tangent line at x=−1.
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Problem 6
A graph of F passes through (2,5), and its tangent there passes through (4,1). Determine F′(2) and the tangent equation. State whether F(4) must equal 1.
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Problem 7
A water-depth function D(t) has D(6)=1.8 meters and D′(6)=−0.12 meters per minute. Explain the meaning of each value and write an equation for the tangent line using y for depth.
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Problem 8
| t (s) | 1.8 | 2.0 | 2.2 |
|---|---|---|---|
| s(t) (m) | 5.24 | 6.00 | 6.84 |
Estimate s′(2) with a centered difference, and explain the units.
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Problem 9
A graph of a smooth function has a tangent at x=3 drawn through (1,−2) and (5,6). Estimate f′(3). Explain why choosing distant points on the curve instead would measure a different rate.
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Problem 10
Use a calculator in radians to estimate f′(1) for f(x)=e^(−x²), using centered differences with h=0.01 and h=0.001. Round the final estimate to six decimals and describe a numerical limitation.
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Problem 11
Decide whether f(x)=|x−2| is continuous and differentiable at 2. Justify both conclusions.
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Problem 12
A graph follows y=x² for x≠1 but has a filled dot at (1,7). Is the represented function differentiable at 1? Explain without calculating a derivative rule.
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Problem 13
For f(x)=x^(1/3), test continuity and differentiability at 0 using the difference quotient. Interpret the tangent geometry.
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Problem 14
Differentiate f(x)=x⁷ and evaluate f′(−1).
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Problem 15
Differentiate f(x)=1/x⁴ and state its domain.
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Problem 16
Differentiate f(x)=x^(3/2) for x>0. Find the right-hand limiting difference quotient at 0 if f(0)=0; explain the distinction between this endpoint result and a two-sided derivative on a real neighborhood.
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Problem 17
Differentiate P(x)=5x⁴−3x²+2x−11.
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Problem 18
Let A(x)=2x³−4x+9 and B(x)=x²−3. Find the derivative of 3A−2B at x=1.
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Problem 19
Differentiate f(x)=3sin x−2cos x+eˣ+ln x, giving the domain of the formula.
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Problem 20
Evaluate lim(h→0)[sin(π/6+h)−sin(π/6)]/h without a calculator.
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Problem 21
Evaluate lim(x→2)[ln x−ln 2]/(x−2), and identify the function and base input.
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Problem 22
Find the tangent line to y=eˣ at x=0.
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Problem 23
Differentiate f(x)=x²cos x.
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Problem 24
| Quantity | Value |
|---|---|
| f(2) | −1 |
| f′(2) | 3 |
| g(2) | 4 |
| g′(2) | −2 |
Find (fg)′(2).
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Problem 25
A rectangular sheet has length L(t) and width W(t). At t=3 seconds, L=5 cm, W=2 cm, L′=−0.4 cm/s and W′=0.6 cm/s. Find the instantaneous area rate.
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Problem 26
Differentiate f(x)=(x²+2)/(x+1), stating the domain.
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Problem 27
| Quantity | Value |
|---|---|
| f(1) | 6 |
| f′(1) | −2 |
| g(1) | 3 |
| g′(1) | 4 |
Find (f/g)′(1).
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Problem 28
Find the derivative of R(x)=(x²+1)/x by first rewriting it. Verify the result with the quotient rule.
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Problem 29
Differentiate f(x)=tan x−cot x and evaluate f′(π/4).
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Problem 30
Differentiate f(x)=sec x+csc x. Evaluate f′(π/6) exactly and state the excluded inputs.
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More chapter problems
Try these problems before opening the solutions. Show your setup, the hypotheses behind your method, and what your answer means.
Derive the complete reciprocal-trig suite
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Worked solution and rubric →Revisit the explanation →Revisit the explanation →Read the relevant lesson →
Diagnose a quotient sign or domain error
Worked solution and rubric →Revisit the explanation →Read the relevant lesson →
Represent the interval rate and its limiting tangent
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Worked solution and rubric →Revisit the explanation →Revisit the explanation →Read the relevant lesson →