AP Unit 3 Problems
Thirty problems for AP Unit 3. Try the problem before opening its separate solution. Write the mathematical setup and the reason your method applies.
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Problem 1
Differentiate f(x)=(2x³−1)⁵.
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Problem 2
Differentiate y=cos(4x²+1).
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Problem 3
Differentiate f(x)=ln(x²+3). State its real domain.
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Problem 4
Differentiate f(x)=e^(sin(3x)).
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Problem 5
| Quantity | Value |
|---|---|
| g(2) | 4 |
| g′(2) | −3 |
| f′(2) | 7 |
| f′(4) | 5 |
The functions are differentiable at the relevant inputs. Find (f∘g)′(2) and explain which table entry is a distractor.
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Problem 6
A sensor converts volume V liters into height H(V) meters. At t=5 minutes, V=8 L and V′=−0.6 L/min. The graph of H has tangent slope 0.25 m/L at V=8. Find the instantaneous height rate.
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Problem 7
For f(x)=ln(1+eˣ), find f′(x) analytically, then use a calculator to evaluate f′(1) to six decimals. Confirm it with a centered numerical difference using h=0.001.
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Problem 8
For x²+y²=25, find dy/dx and the tangent line at (3,4).
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Problem 9
For x²+xy+y²=7, find dy/dx at (1,2).
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Problem 10
For eʸ+x²=1, find dy/dx and the slope at (0,0). State the x-domain of the real y-branch.
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Problem 11
For xy+sin y=1, find the tangent slope at (1,0) if possible. Explain your first check.
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Problem 12
For the circle x²+y²=9, analyze the tangent at (3,0). Does the failure of the formula y′=−x/y imply the curve has no tangent there?
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Problem 13
On the upper branch of x²+xy+y²=7, use a calculator to find y at x=1.2, then find the tangent slope to six decimals. Keep the exact branch formula before rounding.
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Problem 14
| x | f(x) | f′(x) |
|---|---|---|
| 1 | 8 | −2 |
| 2 | 5 | −4 |
| 3 | 0 | −6 |
Use the table to find f⁻¹(5) and (f⁻¹)′(5).
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Problem 15
For f(x)=3x+2, find its inverse g and verify the inverse derivative formula at input 8.
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Problem 16
On x≥0, f(x)=x²+1 has inverse g(x)=√(x−1). Find g′(10) by the inverse derivative rule and confirm with the explicit formula.
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Problem 17
The strictly increasing function f(x)=x³+x has a differentiable inverse g. Use a calculator to solve f(a)=5, then estimate g′(5) to six decimals. Explain why f′(5) is not the needed value.
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Problem 18
For f(x)=x³ with inverse g(x)=x^(1/3), decide whether g′(0) exists as a finite number. Why is 1/f′(0) not a valid numeric answer?
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Problem 19
Differentiate y=arcsin(2x). State its real function domain and the domain of its finite derivative.
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Problem 20
Differentiate y=arccos(x²). Evaluate y′(1/2) exactly and state where the formula is finite.
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Problem 21
Differentiate y=arctan(√x) for x>0. Does this formula give a finite right derivative at x=0?
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Problem 22
Use implicit differentiation to derive the derivative of y=arctan x on its principal branch −π/2<y<π/2.
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Problem 23
Use the branches stated in Topic 3.4. Find (a) d(arccot x)/dx; (b) d(arcsec x)/dx at x=−2; (c) d(arccsc x)/dx at x=2. Include the relevant denominator conditions.
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Problem 24
Choose and apply a coherent procedure to differentiate f(x)=x²sin(3x). Name every rule needed.
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Problem 25
For x>1, differentiate y=ln[(x+1)/(x−1)] using a helpful rewrite, then simplify.
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Problem 26
Differentiate f(x)=e^(x²)/(1+x). Explain the outer rule choice and state the domain.
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Problem 27
For f(x)=x⁵−2x³+7, find f′, f″ and f‴. Evaluate f‴(1).
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Problem 28
A particle has position s(t)=sin(2t) meters, with t in seconds. Find velocity and acceleration and evaluate acceleration at t=π/4 seconds.
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Problem 29
For f(x)=eˣsin x, find f″ and f‴.
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Problem 30
For the upper local branch of x²+y²=25, find y″ at (3,4). Use the differentiated equation rather than treating y as a constant.
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