For , let h(x)= arcsec (2x). A line has slope…

MATHEMATICS · Multi-part · AP · Calculus AB · Differentiation: Composite, Implicit, and Inverse Functions · Differentiation: Composite, Implicit, and Inverse Functions

Problem

For x≥12x\ge \dfrac12, let h(x)=arcsec⁡(2x).h(x)=\operatorname{arcsec}(2x). A line LL has slope 22. (a) Show that h′(x)=2∣2x∣4x2−1h'(x)=\dfrac{2}{|2x|\sqrt{4x^{2}-1}} for x>12x>\dfrac12. (b) Determine all x>12x>\dfrac12 at which the tangent line to the graph of hh is parallel to LL. Give exact values. (c) For one value obtained in part (b), write the equation of the corresponding tangent line. (d) Find the acute angle between this tangent line and the xx-axis, expressing your answer in radians.

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