Consider the trigonometric function: f(x) = tan(x) or y = tan(x). Assume you are

Question

Consider the trigonometric function: f(x) = tan(x) or y = tan(x). Assume you are doing an experiment involving small angles, i.e. near x = 0 radians, and so you need to approximate tangent values for those small angles using Calculus. Construct an equation of the tangent line to this function at x = 0. I am-looking for an exact, concrete equation here: there should only be numbers and the variables x and y in your final, simple equation. You should know the exact values of all trigonometric functions at zero; so, I should see no trig functions in your final equation. Construct the linearization of the function at x = 0, i.e. convert your tangent line equation from Part A into the linear function L(x) as in class. Use your linearization from Part B to approximate tan(pi/96) to six decimal places. Is the approximation from Part C an appropriate one to use? Give your yes or no opinion, and then support your opinion using relevant evidence of your choosing. Construct the differential dy of this function. Compute delta y for x = 0 and delta x = dx = pi/4. Round to six decimal places, if needed. Compute dy for the same values in Part F. Round to six decimal places, if needed. Would dy be an appropriate approximation for delta y here? Give your yes or no opinion, and then support your opinion using relevant evidence of your choosing.

Explanation / Answer

A.
y = tan(x)
when x = 0 , y = tan(0) = 0
so the point through which the tangnt line passes is (0,y(0)) = (0,0)

now the slope of the tangent line at (0,0) is the derivative of y=tanx

=> y' = m = slope = sec^2(x)
y'(0) = sec^2(0) = 1

so now we have a point and the slope so we could use the point slope form to find the equation of
the tangent line
=> (y-y1) = m(x-x1)

y - 0 = 1(x-0)

=> the equation is : y = x

B> the linearization L(x) for the function at x = 0 is

lets use La(x) = linearization of function f at x = 0

La(x) = f(a)+f'(a)(x-a)

L(x) = f(0) + f'(0)(x-0)
L(x) = 0 + 1(x) = x
L(x) = x

Related Questions

Navigate

• Browse (All)
• Browse C
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.