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- https://math.libretexts.org/Bookshelves/Precalculus/Active_Prelude_to_Calculus_(Boelkins)/05%3A_Polynomial_and_Rational_Functions/5.04%3A_Rational_FunctionsStructurally, we observe that \(AV_{[2,2+h]}\) is a ratio of the two functions \(-64h - 16h^2\) and \(h\text{.}\) Moreover, both the numerator and the denominator of the expression are themselves poly...Structurally, we observe that \(AV_{[2,2+h]}\) is a ratio of the two functions \(-64h - 16h^2\) and \(h\text{.}\) Moreover, both the numerator and the denominator of the expression are themselves polynomial functions of the variable \(h\text{.}\) Note that we may be especially interested in what occurs as \(h \to 0\text{,}\) as these values will tell us the average velocity of the moving ball on shorter and shorter time intervals starting at \(t = 2\text{.}\) At the same time, \(AV_{[2,2+h]}\) …
- https://math.libretexts.org/Bookshelves/Precalculus/Active_Prelude_to_Calculus_(Boelkins)/03%3A_Exponential_and_Logarithmic_Functions/3.06%3A_Modeling_temperature_and_populationIn Section 3.2, we learned that Newton's Law of Cooling, which states that an object's temperature changes at a rate proportional to the difference between its own temperature and the surrounding temp...In Section 3.2, we learned that Newton's Law of Cooling, which states that an object's temperature changes at a rate proportional to the difference between its own temperature and the surrounding temperature, results in the object's temperature being modeled by functions of the form \(F(t) = ab^t + c\text{.}\) In light of our subsequent work in Section 3.3 with the natural base \(e\text{,}\) as well as the fact that \(0 \lt b \lt 1\) in this model, we know that Newton's Law of Cooling implies t…
- https://math.libretexts.org/Bookshelves/Precalculus/Active_Prelude_to_Calculus_(Boelkins)/01%3A_Relating_Changing_Quantities/1.01%3A_Changing_in_TandemMathematics is the art of making sense of patterns. One way that patterns arise is when two quantities are changing in tandem. In this setting, we may make sense of the situation by expressing the rel...Mathematics is the art of making sense of patterns. One way that patterns arise is when two quantities are changing in tandem. In this setting, we may make sense of the situation by expressing the relationship between the changing quantities through words, through images, through data, or through a formula.
- https://math.libretexts.org/Bookshelves/Precalculus/Active_Prelude_to_Calculus_(Boelkins)/02%3A_Circular_Functions/2.04%3A_Sinusoidal_FunctionsGiven a function \(y = f(t)\) and a real number \(k \gt 0\text{,}\) the transformed function \(y = h(t) = f(kt)\) is a horizontal stretch of the graph of \(f\text{.}\) Every point \((t,f(t))\) on the ...Given a function \(y = f(t)\) and a real number \(k \gt 0\text{,}\) the transformed function \(y = h(t) = f(kt)\) is a horizontal stretch of the graph of \(f\text{.}\) Every point \((t,f(t))\) on the graph of \(f\) gets stretched horizontally to the corresponding point \((\frac{1}{k}t,f(t))\) on the graph of \(h\text{.}\) If \(0 \lt k \lt 1\text{,}\) the graph of \(h\) is a stretch of \(f\) away from the \(y\)-axis by a factor of \(\frac{1}{k}\text{;}\) if \(k \gt 1\text{,}\) the graph of \(h\)…
- https://math.libretexts.org/Bookshelves/Precalculus/Active_Prelude_to_Calculus_(Boelkins)/zz%3A_Back_Matter/10%3A_Index
- https://math.libretexts.org/Bookshelves/Precalculus/Active_Prelude_to_Calculus_(Boelkins)/02%3A_Circular_Functions
- https://math.libretexts.org/Bookshelves/Precalculus/Active_Prelude_to_Calculus_(Boelkins)/05%3A_Polynomial_and_Rational_Functions/5.02%3A_PolynomialsIf \((x-r)^n\) is a factor of a polynomial \(p\text{,}\) then \(x = r\) is a zero of \(p\) of multiplicity \(n\text{,}\) and near \(x = r\) the graph of \(p\) looks like either \(-x^n\) or \(x^n\) doe...If \((x-r)^n\) is a factor of a polynomial \(p\text{,}\) then \(x = r\) is a zero of \(p\) of multiplicity \(n\text{,}\) and near \(x = r\) the graph of \(p\) looks like either \(-x^n\) or \(x^n\) does near \(x = 0\text{.}\) That is, the shape of the graph near the zero is determined by the multiplicity of the zero.
- https://math.libretexts.org/Bookshelves/Precalculus/Active_Prelude_to_Calculus_(Boelkins)/01%3A_Relating_Changing_Quantities/1.02%3A_Functions-_Modeling_RelationshipsIt's possible to show that the formula for the function \(g\) is \(g(t) = \left( \frac{t}{\pi} \right)^{1/3}\text{.}\) Use a computational device to generate two plots: on the axes at left, the graph ...It's possible to show that the formula for the function \(g\) is \(g(t) = \left( \frac{t}{\pi} \right)^{1/3}\text{.}\) Use a computational device to generate two plots: on the axes at left, the graph of the model \(h = g(t) = \left( \frac{t}{\pi} \right)^{1/3}\) on the domain that you decided in (c); on the axes at right, the graph of the abstract function \(y = p(x) = \left( \frac{t}{\pi} \right)^{1/3}\) on a wider domain than that of \(g\text{.}\) What are the domain and range of \(p\) and ho…
- https://math.libretexts.org/Bookshelves/Precalculus/Active_Prelude_to_Calculus_(Boelkins)/04%3A_Trigonometry/4.01%3A_Right_trianglesIf we let the angle formed by the hypotenuse and the horizontal leg have measure \(\theta\text{,}\) then the right triangle with hypotenuse \(1\) has horizontal leg of length \(\cos(\theta)\) and vert...If we let the angle formed by the hypotenuse and the horizontal leg have measure \(\theta\text{,}\) then the right triangle with hypotenuse \(1\) has horizontal leg of length \(\cos(\theta)\) and vertical leg of length \(\sin(\theta)\text{.}\) If we consider now consider a similar right triangle with hypotenuse of length \(r \ne 1\text{,}\) we can view that triangle as a magnification of a triangle with hypotenuse \(1\text{.}\) These observations, combined with our work in Activity \(\PageIndex…
- https://math.libretexts.org/Bookshelves/Precalculus/Active_Prelude_to_Calculus_(Boelkins)/01%3A_Relating_Changing_Quantities/1.03%3A_The_Average_Rate_of_Change_of_a_FunctionFor example, in Preview Activity \(\PageIndex{1}\), the units on \(AV_{[1.5,2.5]} = -32\) are “feet per second” since the units on the numerator are “feet” and on the denominator “seconds”. Morever, \...For example, in Preview Activity \(\PageIndex{1}\), the units on \(AV_{[1.5,2.5]} = -32\) are “feet per second” since the units on the numerator are “feet” and on the denominator “seconds”. Morever, \(-32\) is numerically the same value as the slope of the line that connects the two corresponding points on the graph of the position function, as seen in Figure \(\PageIndex{2}\) The fact that the average rate of change is negative in this example indicates that the ball is falling.
- https://math.libretexts.org/Bookshelves/Precalculus/Active_Prelude_to_Calculus_(Boelkins)/03%3A_Exponential_and_Logarithmic_Functions/3.01%3A_Exponential_Growth_and_DecayFrom the data in Table \(\PageIndex{10}\), we see that the average rate of change is increasing as we increase the value of \(t\text{.}\) We naturally say that \(f\) appears to be “increasing at an in...From the data in Table \(\PageIndex{10}\), we see that the average rate of change is increasing as we increase the value of \(t\text{.}\) We naturally say that \(f\) appears to be “increasing at an increasing rate”. For the function \(g\text{,}\) we first notice that its average rate of change is always negative, but also that the average rate of change gets less negative as we increase the value of \(t\text{.}\) Said differently, the average rate of change of \(g\) is also increasing as we inc…
