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Calculus | Packer Collegiate Institute

FURTHERING our understanding of Basic Derivatives

1(a). Carefully sketch the graph of and . Find the value(s) of at which the graphs have parallel tangent lines.

1(b). Using your graphing calculators, graph and . Find the value(s) of at which the graphs have parallel tangent lines.

2 (a). Determine coefficients and such that satisfies and .

2 (b). Determine coefficients , and such that satisfies , , and .

3. Biologists have observed that the pulse rate (in beats per minute) in animals is related to body mass (in kilograms) by the approximate formula . This is one of many allometric scaling laws prevalent in biology. (a). In general, the smaller the body mass of an animal, the ______________ the pulse rate of the animal. In general, the larger the body mass of an animal, the ______________ the pulse rate of the animal. (b). Is the absolute value increasing or decreasing as increases?

Mass (kg)

Pulse (beats/min)

Guinea Pig

Goat

Man

Cattle

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(c). Find an equation of the tangent line at the points on figure that represent goat () and man ( ).

4 (a). Find the equation of the tangent line to the hyperbola (alternatively written

) at . Sketch the hyperbola and this tangent line. Find the area of the

triangle bounded by the tangent line, the x-axis, and the y-axis. 4 (b). Find the equation of the tangent line to the hyperbola (alternatively written

) at . Sketch the hyperbola and this tangent line. Find the area of the

triangle bounded by the tangent line, the x-axis, and the y-axis.4 (c). Find the equation of the tangent line to the hyperbola (alternatively written

) at . Find the area of the triangle bounded by the tangent line, the x-axis,

and the y-axis.

5. The curve is called the witch of Agnesi after the Italian mathematician Maria

Agnesi (1718-1799) who wrote one of the first books on calculus. This strange name is the result of a mistranslation of the Italian word la versiera meaning “that which turns.” Find the equations of the tangent lines at and .

6 (a). Calculate the derivative of .

(b). Calculate the derivative of . [Hint: rewrite the denominator as .]

7 (a). You learned that you can find an approximate derivative by picking two points really close to each other and finding the slope between them. Using this approximate method, find the derivative of the following functions at . Round your answer to 3 decimal places.

(b). Approximate the following natural logarithms using your calculator, rounding to 3 decimal places.

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