Tangent Tactics Worksheet
AP Calculus AB / BC • Unit 2: Differentiation
Tangent Tactics Worksheet
Rules & Tangents
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Score: / 30
1
Derivative Rules Mastery
Find the first derivative \(f'(x)\) or \(y'\). Simplify fully.
1. Combined Power & Trig 3 pts
\(f(x) = 5x^4 - 2\sqrt{x} + 4\cos(x) - \frac{3}{x^2}\)
\(f'(x) =\)
2. Product Rule 3 pts
\(g(x) = (3x^2 - 5x) \cdot e^x\)
\(g'(x) =\)
3. Quotient Rule 3 pts
\(h(x) = \frac{2x - 1}{x^2 + 3}\)
\(h'(x) =\)
4. Chain Rule 3 pts
\(k(x) = \sin^3(4x^2 + 1)\)
\(k'(x) =\)
2
Tangent Lines & Geometric Slopes
Use derivative evaluations to determine exact equations.
5. Tangent & Normal Equations 4 pts
Given \(f(x) = x^3 - 4x + 3\):
(a) Find the slope of the tangent line at \(x = 2\).
(b) Write the point-slope equation of the tangent line.
(c) State the slope of the line perpendicular (normal) at this point.
Tangent Eq: \(y - \quad\quad = \quad\quad(x - 2)\)
Normal Slope: \(m_{\perp} =\)
6. Horizontal Tangents 4 pts
Determine all coordinate points \((x, y)\) on the curve:
\(y = 2x^3 - 3x^2 - 12x + 8\)
where the tangent line is horizontal (\(m = 0\)). Show all algebraic solving steps.
Points: \(( \quad\quad, \quad\quad )\) and \(( \quad\quad, \quad\quad )\)
Tangent Tactics • Differential Calculus Practice Turn page for Particle Motion & Challenge Synthesis →
Rates of Change & Motion Analysis
Unit 2 Applications • Derivative as Instantaneous Rate
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Rectilinear Particle Motion
7 Points Total
A particle moves along the \(x\)-axis such that its position in meters at time \(t \ge 0\) seconds is modeled by:
\(s(t) = \frac{1}{3}t^3 - 3t^2 + 8t + 1\)
(a) Velocity & Acceleration Functions (2 pts)
Find symbolic expressions for \(v(t) = s'(t)\) and \(a(t) = v'(t)\).
\(v(t) =\)
\(a(t) =\)
(b) Times at Rest (1 pt)
At what value(s) of \(t\) is the particle momentarily at rest?
At rest at \(t =\) seconds
(c) Direction of Motion (2 pts)
On what open interval(s) of \(t\) is the particle moving to the left? Justify.
Interval:
(d) Speed: Increasing or Decreasing? (2 pts)
At \(t = 5\), is the speed increasing or decreasing? Justify with \(v(5)\) and \(a(5)\).
Conclusion:
4
Synthesis & Applied Rates
6 Points Total
8. Tangency Parameter 3 pts
Find the value of the constant \(c\) such that the line \(y = 4x + c\) is tangent to the curve:
\(f(x) = x^2 - 2x + 5\)
Also determine the exact coordinates \((x_0, y_0)\) of tangency.
Point: \(( \quad\quad, \quad\quad )\) \(c =\)
9. Draining Conical Tank 3 pts
The volume of oil in gallons remaining in a tank after \(t\) hours (\(0 \le t \le 10\)) is:
\(V(t) = 200\left(1 - \frac{t}{10}\right)^2\)
(a) Find \(V'(4)\) with units.
(b) Explain the physical meaning of your result.
\(V'(4) =\) units:
Self-Check: Did you label all units in rate problems and simplify all rational expressions?
Units included Derivatives verified
