By Frank Ayres; Elliot Mendelson
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2. Page 29 Solution. 2. 44. 2 Differentiate Solution. 3 Differentiate Solution. 4 Differentiate Page 30 Solution. 5 Differentiate Solution. 6 Given fix) = 1 - x3, find f'(-4) and f'(4). Solution. 7 Differentiate the function: Page 31 Solution. 8 Determine the rate of change of the area of a circle with respect to its radius, R. Also, evaluate the rate of change when R=5. Solution. The area of a circle is related to the radius by the function: Therefore, the rate of change of the area of the circle in terms of the radius, R, is which is the circumference of a circle.
The function has a minimum (= 0) when x = 0. Concavity An arc of a curve y = f(x) is called concave upward if, at each of its points, the arc lies above the tangent at that point. As x increases, f'(x) either is of the same sign and increasing (as on the interval b < x < s of Figure 3-3) or changes sign from negative to positive (as on the interval c < x < u). In either case, the slope f'(x) is increasing hence f"(x) > 0. An arc of a curve y = f(x) is called concave downward if, at each of its points, the arc lies below the tangent at that point.
And, since a full circle has a circumference of 2π rad, we can write 1 rad = 180/π degrees and 1° = π/180 rad. ) Suppose AOB is measured as α degrees; then we formulate the arc length and area of the sector as: Suppose next that AOB is measured as θ rad; then Page 55 Note! A comparison of Eqs. 2) will make clear one of the advantages of radian measure. , a real number. Trigonometric Functions Let θ be any real number. Construct the angle whose measure is θ radians with vertex at the origin of a rectangular coordinate system, and initial side along the positive x axis (see Figure 4-2).
Calculus Crash Course by Frank Ayres; Elliot Mendelson