Consider the scalar function and the vector–valued function for .

MATHEMATICS · Multi-part · AP · Calculus BC · Parametric, Polar, and Vector-Valued Functions · Parametric, Polar, and Vector-Valued Functions

Problem

Consider the scalar function f(t)=e3tf(t)=e^{3t} and the vector–valued function r(t)=(t2, sin⁡t, ln⁡(1+t))\mathbf r(t)=\bigl(t^2,\,\sin t,\,\ln(1+t)\bigr) for t>−1t> -1. (a) Calculate the derivative ddt[f(t) r(t)]\dfrac{d}{dt}\bigl[f(t)\,\mathbf r(t)\bigr]. Express your answer as a single vector whose components are simplified functions of tt. (b) Evaluate the derivative found in part (a) at t=0t=0 and determine its magnitude. Provide exact values, not decimal approximations. (c) Explain how the scalar factor f(t)f(t) affects both the direction and the magnitude of the derivative you computed in part (a). Support your explanation with reference to the specific expressions obtained.

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