10. Tangent line for exponential and trigonometric functions a) Find the tangent line to the graph...

 

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10. Tangent line for exponential and trigonometric functions a) Find the tangent line to the graph of f(x) at the point (1, f(1) where f(x) is given as follows f(x) = x2 + e* + 1 { f(x) = x^2 + e^x + 1 } b) Find the tangent line to the graph of g(x) at the point (/3, g(n/3)) where g(x) is given as follows (pi)/3, g((pi)/3) g(x) = 2 sin x - cos x ( g(x) = 2 sin x - cos x } %3D QUESTION 11 Position, Velocity, and Acceleration (Trigonometric) Example of Simple Harmonic Motion The position function r(t) for an object under simple harmonic motion is given by the following expression r(t) = 5 cos t { r(t) = 5 cost } a) Find the velocity function v(t) = r'(t) and the acceleration function a(t) = v'(t) = r"(t) b) Show that acceleration a(t) is the negative of position (this corresponds to a force that equals the displacement, but in the opposite direction} Try writing down this relation between r(t) and r"(t) in an equation (since this equation involves a function r(t) and its (second) derivative r"(t), it is called a differential equation c) Show that the same relation holds for the function R(t) given below, a similar function involving both sine and cosine R(t) = 3 sin t + 4 cos t %3D QUESTION 12 Product Rule Apply the product rule to find the derivative of each of the following functions. a) f(x) = x tan x { f(x) = x tan x } b) g(x) = (2x2 + 3x + 1) e* %3D { g(x) = (2x^2 + 3x + 1) e^x 10. Tangent line for exponential and trigonometric functions a) Find the tangent line to the graph of f(x) at the point (1, f(1) where f(x) is given as follows f(x) = x2 + e* + 1 { f(x) = x^2 + e^x + 1 } b) Find the tangent line to the graph of g(x) at the point (/3, g(n/3)) where g(x) is given as follows (pi)/3, g((pi)/3) g(x) = 2 sin x - cos x ( g(x) = 2 sin x - cos x } %3D QUESTION 11 Position, Velocity, and Acceleration (Trigonometric) Example of Simple Harmonic Motion The position function r(t) for an object under simple harmonic motion is given by the following expression r(t) = 5 cos t { r(t) = 5 cost } a) Find the velocity function v(t) = r'(t) and the acceleration function a(t) = v'(t) = r"(t) b) Show that acceleration a(t) is the negative of position (this corresponds to a force that equals the displacement, but in the opposite direction} Try writing down this relation between r(t) and r"(t) in an equation (since this equation involves a function r(t) and its (second) derivative r"(t), it is called a differential equation c) Show that the same relation holds for the function R(t) given below, a similar function involving both sine and cosine R(t) = 3 sin t + 4 cos t %3D QUESTION 12 Product Rule Apply the product rule to find the derivative of each of the following functions. a) f(x) = x tan x { f(x) = x tan x } b) g(x) = (2x2 + 3x + 1) e* %3D { g(x) = (2x^2 + 3x + 1) e^x

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