Solving Optimization Problems - AP Calculus BC
Card 1 of 30
Calculate the derivative: $f(x) = 5x^4$.
Calculate the derivative: $f(x) = 5x^4$.
Tap to reveal answer
$f'(x) = 20x^3$. Power rule: $4 \times 5x^{4-1} = 20x^3$.
$f'(x) = 20x^3$. Power rule: $4 \times 5x^{4-1} = 20x^3$.
← Didn't Know|Knew It →
State the method to solve optimization problems with constraints.
State the method to solve optimization problems with constraints.
Tap to reveal answer
Use Lagrange multipliers. Technique for handling equality constraints.
Use Lagrange multipliers. Technique for handling equality constraints.
← Didn't Know|Knew It →
What is the derivative of $f(x) = \text{e}^{2x}$?
What is the derivative of $f(x) = \text{e}^{2x}$?
Tap to reveal answer
$f'(x) = 2\text{e}^{2x}$. Chain rule applied to exponential with coefficient.
$f'(x) = 2\text{e}^{2x}$. Chain rule applied to exponential with coefficient.
← Didn't Know|Knew It →
Identify the constraint in maximizing the volume of a box.
Identify the constraint in maximizing the volume of a box.
Tap to reveal answer
Surface area or material limits. Material availability restricts the design space.
Surface area or material limits. Material availability restricts the design space.
← Didn't Know|Knew It →
Calculate the derivative: $f(x) = \text{tan}(x)$.
Calculate the derivative: $f(x) = \text{tan}(x)$.
Tap to reveal answer
$f'(x) = \text{sec}^2(x)$. Standard derivative of tangent function.
$f'(x) = \text{sec}^2(x)$. Standard derivative of tangent function.
← Didn't Know|Knew It →
Find the derivative of $g(x) = \frac{1}{x}$.
Find the derivative of $g(x) = \frac{1}{x}$.
Tap to reveal answer
$g'(x) = -\frac{1}{x^2}$. Rewrite as $x^{-1}$ and apply power rule.
$g'(x) = -\frac{1}{x^2}$. Rewrite as $x^{-1}$ and apply power rule.
← Didn't Know|Knew It →
What step follows finding the critical points in optimization?
What step follows finding the critical points in optimization?
Tap to reveal answer
Evaluate the objective function at critical points. Determines which critical point gives optimal value.
Evaluate the objective function at critical points. Determines which critical point gives optimal value.
← Didn't Know|Knew It →
Identify the formula for the derivative of $f(x) = x^2 + 3x$.
Identify the formula for the derivative of $f(x) = x^2 + 3x$.
Tap to reveal answer
$f'(x) = 2x + 3$. Power rule applied: bring down exponent, reduce by 1.
$f'(x) = 2x + 3$. Power rule applied: bring down exponent, reduce by 1.
← Didn't Know|Knew It →
State the condition for a relative maximum using $f''(x)$.
State the condition for a relative maximum using $f''(x)$.
Tap to reveal answer
$f''(x) < 0$ at critical point. Negative second derivative indicates downward concavity.
$f''(x) < 0$ at critical point. Negative second derivative indicates downward concavity.
← Didn't Know|Knew It →
Determine the derivative: $f(x) = x^3 - 3x^2 + 2$.
Determine the derivative: $f(x) = x^3 - 3x^2 + 2$.
Tap to reveal answer
$f'(x) = 3x^2 - 6x$. Power rule applied to each term separately.
$f'(x) = 3x^2 - 6x$. Power rule applied to each term separately.
← Didn't Know|Knew It →
Determine the derivative: $f(x) = 3x^{-2}$.
Determine the derivative: $f(x) = 3x^{-2}$.
Tap to reveal answer
$f'(x) = -6x^{-3}$. Power rule: $3 \times (-2)x^{-3} = -6x^{-3}$.
$f'(x) = -6x^{-3}$. Power rule: $3 \times (-2)x^{-3} = -6x^{-3}$.
← Didn't Know|Knew It →
State the condition for a relative minimum using $f''(x)$.
State the condition for a relative minimum using $f''(x)$.
Tap to reveal answer
$f''(x) > 0$ at critical point. Positive second derivative indicates upward concavity.
$f''(x) > 0$ at critical point. Positive second derivative indicates upward concavity.
← Didn't Know|Knew It →
Identify the derivative of $f(x) = x^{\frac{1}{2}}$.
Identify the derivative of $f(x) = x^{\frac{1}{2}}$.
Tap to reveal answer
$f'(x) = \frac{1}{2}x^{-\frac{1}{2}}$. Power rule with fractional exponent: $\frac{1}{2}x^{-1/2}$.
$f'(x) = \frac{1}{2}x^{-\frac{1}{2}}$. Power rule with fractional exponent: $\frac{1}{2}x^{-1/2}$.
← Didn't Know|Knew It →
State the critical points condition for $f'(x) = 0$.
State the critical points condition for $f'(x) = 0$.
Tap to reveal answer
Critical points occur where $f'(x) = 0$ or is undefined. These are potential locations for extrema.
Critical points occur where $f'(x) = 0$ or is undefined. These are potential locations for extrema.
← Didn't Know|Knew It →
What does $f'(x) = 0$ imply about $f(x)$?
What does $f'(x) = 0$ imply about $f(x)$?
Tap to reveal answer
Possible extremum or inflection point. Zero slope indicates potential maximum or minimum.
Possible extremum or inflection point. Zero slope indicates potential maximum or minimum.
← Didn't Know|Knew It →
Calculate the derivative: $f(x) = e^x$.
Calculate the derivative: $f(x) = e^x$.
Tap to reveal answer
$f'(x) = e^x$. Exponential function is its own derivative.
$f'(x) = e^x$. Exponential function is its own derivative.
← Didn't Know|Knew It →
What is the derivative of $f(x) = \text{ln}(x)$?
What is the derivative of $f(x) = \text{ln}(x)$?
Tap to reveal answer
$f'(x) = \frac{1}{x}$. Standard derivative of natural logarithm function.
$f'(x) = \frac{1}{x}$. Standard derivative of natural logarithm function.
← Didn't Know|Knew It →
Find the derivative of $f(x) = \text{arcsin}(x)$.
Find the derivative of $f(x) = \text{arcsin}(x)$.
Tap to reveal answer
$f'(x) = \frac{1}{\text{√}(1-x^2)}$. Standard derivative of inverse sine function.
$f'(x) = \frac{1}{\text{√}(1-x^2)}$. Standard derivative of inverse sine function.
← Didn't Know|Knew It →
What is the derivative of $f(x) = \text{log}_a(x)$?
What is the derivative of $f(x) = \text{log}_a(x)$?
Tap to reveal answer
$f'(x) = \frac{1}{x \text{ln}(a)}$. General formula for logarithm base $a$ derivative.
$f'(x) = \frac{1}{x \text{ln}(a)}$. General formula for logarithm base $a$ derivative.
← Didn't Know|Knew It →
State the chain rule for derivatives.
State the chain rule for derivatives.
Tap to reveal answer
If $y = g(f(x))$, then $dy/dx = g'(f(x))f'(x)$. Rule for differentiating composite functions.
If $y = g(f(x))$, then $dy/dx = g'(f(x))f'(x)$. Rule for differentiating composite functions.
← Didn't Know|Knew It →
What is the condition for an inflection point?
What is the condition for an inflection point?
Tap to reveal answer
$f''(x) = 0$ and $f''(x)$ changes sign. Point where concavity changes direction.
$f''(x) = 0$ and $f''(x)$ changes sign. Point where concavity changes direction.
← Didn't Know|Knew It →
Calculate the derivative: $f(x) = \text{cos}(x)$.
Calculate the derivative: $f(x) = \text{cos}(x)$.
Tap to reveal answer
$f'(x) = -\text{sin}(x)$. Derivative of cosine is negative sine.
$f'(x) = -\text{sin}(x)$. Derivative of cosine is negative sine.
← Didn't Know|Knew It →
Find the derivative of $f(x) = \frac{x^2}{2}$.
Find the derivative of $f(x) = \frac{x^2}{2}$.
Tap to reveal answer
$f'(x) = x$. Coefficient $\frac{1}{2}$ remains, power rule on $x^2$.
$f'(x) = x$. Coefficient $\frac{1}{2}$ remains, power rule on $x^2$.
← Didn't Know|Knew It →
What is the purpose of Lagrange multipliers?
What is the purpose of Lagrange multipliers?
Tap to reveal answer
To find extrema of functions subject to constraints. Method for optimization with equality constraints.
To find extrema of functions subject to constraints. Method for optimization with equality constraints.
← Didn't Know|Knew It →
Find the critical points of $h(x) = x^2 - 4x + 4$.
Find the critical points of $h(x) = x^2 - 4x + 4$.
Tap to reveal answer
$x = 2$. Set $h'(x) = 2x - 4 = 0$, solve for $x$.
$x = 2$. Set $h'(x) = 2x - 4 = 0$, solve for $x$.
← Didn't Know|Knew It →
What is the derivative of a constant function $f(x) = c$?
What is the derivative of a constant function $f(x) = c$?
Tap to reveal answer
$f'(x) = 0$. Constants have zero rate of change.
$f'(x) = 0$. Constants have zero rate of change.
← Didn't Know|Knew It →
What is the derivative of $f(x) = \text{sin}(x)$?
What is the derivative of $f(x) = \text{sin}(x)$?
Tap to reveal answer
$f'(x) = \text{cos}(x)$. Standard trigonometric derivative rule.
$f'(x) = \text{cos}(x)$. Standard trigonometric derivative rule.
← Didn't Know|Knew It →
Find the critical points of $h(x) = x^2 - 4x + 4$.
Find the critical points of $h(x) = x^2 - 4x + 4$.
Tap to reveal answer
$x = 2$. Set $h'(x) = 2x - 4 = 0$, solve for $x$.
$x = 2$. Set $h'(x) = 2x - 4 = 0$, solve for $x$.
← Didn't Know|Knew It →
Calculate the derivative: $f(x) = 5x^4$.
Calculate the derivative: $f(x) = 5x^4$.
Tap to reveal answer
$f'(x) = 20x^3$. Power rule: $4 \times 5x^{4-1} = 20x^3$.
$f'(x) = 20x^3$. Power rule: $4 \times 5x^{4-1} = 20x^3$.
← Didn't Know|Knew It →
What is the derivative of $f(x) = \text{sin}(x)$?
What is the derivative of $f(x) = \text{sin}(x)$?
Tap to reveal answer
$f'(x) = \text{cos}(x)$. Standard trigonometric derivative rule.
$f'(x) = \text{cos}(x)$. Standard trigonometric derivative rule.
← Didn't Know|Knew It →