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- https://math.libretexts.org/Bookshelves/Calculus/Informal_Calculus_with_Applications_to_Biological_and_Environmental_Sciences_(Seacrest)/00%3A_Front_Matter
- https://math.libretexts.org/Bookshelves/Calculus/Informal_Calculus_with_Applications_to_Biological_and_Environmental_Sciences_(Seacrest)/04%3A_More_Derivative_Intuition/4.05%3A_OptimizationThe high point is called a maximum, and the low point is called a minimum. Optimization The height of a baseball follows the function \(h(t) = -5t^2 + 20t + 10\), where \(h\) is measured in meters and...The high point is called a maximum, and the low point is called a minimum. Optimization The height of a baseball follows the function \(h(t) = -5t^2 + 20t + 10\), where \(h\) is measured in meters and \(t\) is measured in seconds.. What value of \(t\) maximizes the height? We will follow the maxim “optimization happens when the derivative is zero”. First we find the derivative using the power rule \(h'(t) = -10t + 20\).
- https://math.libretexts.org/Bookshelves/Calculus/Informal_Calculus_with_Applications_to_Biological_and_Environmental_Sciences_(Seacrest)/05%3A_Differential_Equations/5.01%3A_Recurrance_Relations\[\begin{align*} f^0_{t+1} & = 5.23 f^3_t + 18.0 f^4_t + 24.55 f^{5+}_t \\ f^1_{t+1} & = 0.277 f^0_t \\ f^2_{t+1} & = 0.3405 f^1_t \\ f^3_{t+1} & = 0.4675 f^2_t \\ f^4_{t+1} & = 0.4675 f^3_t \\ f^{5+}...\[\begin{align*} f^0_{t+1} & = 5.23 f^3_t + 18.0 f^4_t + 24.55 f^{5+}_t \\ f^1_{t+1} & = 0.277 f^0_t \\ f^2_{t+1} & = 0.3405 f^1_t \\ f^3_{t+1} & = 0.4675 f^2_t \\ f^4_{t+1} & = 0.4675 f^3_t \\ f^{5+}_{t+1} & = 0.4675 f^4_t + 0.4675 f^{5+}_t \\ \end{align*}\] The equation \(f^{5+}_{t+1} = 0.4675 f^4_t + 0.4675 f^{5+}_t\) is a bit more complicated, since trout of age 5+ come from the age 4 trout, but also the age 5+ stay within that category, so there are two ways to get there.
- https://math.libretexts.org/Bookshelves/Calculus/Informal_Calculus_with_Applications_to_Biological_and_Environmental_Sciences_(Seacrest)/03%3A_Rules_for_Derivatives/3.04%3A_Algebra_Tips_and_Tricks_Part_VI_(Logarithms)\[\begin{align*} e^{\ln(x)} & = x \\ \ln(e^x) & = x \\ \ln(x) + \ln(y) & = \ln(xy) \\ \ln(x) - \ln(y) & = \ln\left(\frac{x}{y}\right) \\ a \ln(x) & = \ln(x^a) \end{align*}\] \[\begin{align*} \ln\left(...\[\begin{align*} e^{\ln(x)} & = x \\ \ln(e^x) & = x \\ \ln(x) + \ln(y) & = \ln(xy) \\ \ln(x) - \ln(y) & = \ln\left(\frac{x}{y}\right) \\ a \ln(x) & = \ln(x^a) \end{align*}\] \[\begin{align*} \ln\left(\frac{\sqrt{x} y}{z^3}\right) - \ln\left(\frac{z}{\sqrt{x} y^3}\right) & = \ln(\sqrt{x}) + \ln(y) - \ln(z^3) - [\ln(z) - \ln(\sqrt{x}) - \ln(y^3)] \\ & = \frac{1}{2} \ln(x) + \ln(y) - 3 \ln(z) - \ln(z) + \frac{1}{2} \ln(x) + 3 \ln(y) \\ & = \boxed{\ln(x) + 4 \ln(y) - 4 \ln(z)}. \end{align*}\]
- https://math.libretexts.org/Bookshelves/Calculus/Informal_Calculus_with_Applications_to_Biological_and_Environmental_Sciences_(Seacrest)/02%3A_Derivative_Introduction/2.12%3A_Algebra_Tips_and_Tricks_IV_(Tips_for_dealing_with_fractions)\[\begin{align*} \cfrac{\cfrac{1}{x+1} - \cfrac{1}{x-1}}{\cfrac{1}{x}} \cdot \frac{x(x-1)(x+1)}{x(x-1)(x+1)} & = \cfrac{\cfrac{x(x+1)(x-1)}{x+1} - \cfrac{x(x+1)(x-1)}{x-1}}{\cfrac{x(x+1)(x-1)}{x}}\\ &...\[\begin{align*} \cfrac{\cfrac{1}{x+1} - \cfrac{1}{x-1}}{\cfrac{1}{x}} \cdot \frac{x(x-1)(x+1)}{x(x-1)(x+1)} & = \cfrac{\cfrac{x(x+1)(x-1)}{x+1} - \cfrac{x(x+1)(x-1)}{x-1}}{\cfrac{x(x+1)(x-1)}{x}}\\ & = \cfrac{x(x-1) - x(x+1)}{(x-1)(x+1)}\\ & = \cfrac{x^2 - x - (x^2 + x)}{x^2 - 1} \\ & = \cfrac{x^2 - x - x^2 - x}{x^2 - 1} \\ & = \cfrac{-2x}{x^2 - 1} \end{align*}\]
- https://math.libretexts.org/Bookshelves/Calculus/Informal_Calculus_with_Applications_to_Biological_and_Environmental_Sciences_(Seacrest)/03%3A_Rules_for_Derivatives/3.03%3A_Homework-_Power_RuleCompute the following derivatives. Do not use the definition of the derivative. Instead, use the linearity and power rules we talked about in this section. \(\cfrac{d}{dx}\ x^{15}\) \(15 x^{14}\) \(\c...Compute the following derivatives. Do not use the definition of the derivative. Instead, use the linearity and power rules we talked about in this section. \(\cfrac{d}{dx}\ x^{15}\) \(15 x^{14}\) \(\cfrac{d}{dx}\ 3x^6\) \(\cfrac{d}{dx}\ \cfrac{1}{2}x^{4}\) \(2x^3\) \(\cfrac{d}{dx}\ 3 x^2 + 6x - 1\) \(\cfrac{d}{dx}\ (2x + 3)^2\) \(\cfrac{d}{dx}\ 7 x^{-4}\) \(\cfrac{d}{dx}\ \sqrt{x}\) \(\cfrac{d}{dx}\ \frac{1}{x}\) \(-x^{-2}\) \(\cfrac{d}{dx}\ \sqrt[3]{x^2}\) \(\cfrac{d}{dx}\ \frac{2}{\sqrt{x}}\)
- https://math.libretexts.org/Bookshelves/Calculus/Informal_Calculus_with_Applications_to_Biological_and_Environmental_Sciences_(Seacrest)/07%3A_Rules_for_Integration
- https://math.libretexts.org/Bookshelves/Calculus/Informal_Calculus_with_Applications_to_Biological_and_Environmental_Sciences_(Seacrest)/01%3A_Algebra_Tips_and_Tricks-_Part_I
- https://math.libretexts.org/Bookshelves/Calculus/Informal_Calculus_with_Applications_to_Biological_and_Environmental_Sciences_(Seacrest)/04%3A_More_Derivative_Intuition
- https://math.libretexts.org/Bookshelves/Calculus/Informal_Calculus_with_Applications_to_Biological_and_Environmental_Sciences_(Seacrest)/02%3A_Derivative_Introduction/2.15%3A_Homework-_Examples_of_the_Definition_of_the_Derivative\(\cfrac{\cfrac{1}{x} - \cfrac{1}{x+1}}{\cfrac{1}{x} + \cfrac{1}{x+1}}\) In each case, use the definition of the derivative to find \(f'(x)\) (in other words, take the derivative!) In each case, use t...\(\cfrac{\cfrac{1}{x} - \cfrac{1}{x+1}}{\cfrac{1}{x} + \cfrac{1}{x+1}}\) In each case, use the definition of the derivative to find \(f'(x)\) (in other words, take the derivative!) In each case, use the definition of the derivative to find \(f'(x)\) (in other words, take the derivative!). Recall the derivative of \(f(x) = x^2\) is given by \(2x\). Show that the derivative of \(g(x) = x^2+1\) is \(2x\) using the definition of the derivative.
- https://math.libretexts.org/Bookshelves/Calculus/Informal_Calculus_with_Applications_to_Biological_and_Environmental_Sciences_(Seacrest)/04%3A_More_Derivative_Intuition/4.06%3A_Homework-_OptimizationLocal max at \(x = 0.75\), local min at \(x = 2\), global max at \(x = 2.5\), global min at \(x = 0\). Local min at \(x = -3\), local max at \(x = -2\), local min at \(x = 0\), local max at \(x = 3\) ...Local max at \(x = 0.75\), local min at \(x = 2\), global max at \(x = 2.5\), global min at \(x = 0\). Local min at \(x = -3\), local max at \(x = -2\), local min at \(x = 0\), local max at \(x = 3\) Using a chemotherapy drug on a petri-dish of cancer cells, it is found that \(P(x)\) percent more of the cancer cells are killed using \(x\) milligrams of drug per square centimeter than healthy cells, where \(x\) ranges from \(0\) to \(4\).