Consider the curve f(x) = 5x ? 4. Find the slope of the line connecting the poin
ID: 3215177 • Letter: C
Question
Consider the curve f(x) = 5x ? 4. Find the slope of the line connecting the points. Sketch the curve and those secant lines. (a) (7, f(7)) and (7.2, f(7.2)) (b) (7, f(7)) and (6.8, f(6.8)) (c) Find the slope of the line through (7, f(7)) and (7 + h, f(7 + h)). (d) Consider what happens to that expression as h gets closer and closer to 0. lim h?0 m(h) = ? What is the slope of the tangent line to the curve at x = 7? Sketch the tangent line. Write an equation of the tangent line to the curve at x = 7. (Let x be the independent variable and y be the dependent variable.)Explanation / Answer
first off you need to find the derivative of your equation. The derivative generates the slopes of your tangent lines at any given point. I am going to assume that you have been taught how to do this the fast way (which is multiplying the coefficient by the exponent, then replacing the exponent with 1 minus the exponent. so the derivative of x^4 would be 4x^3, or the derivative of (X/3)^4 would be ((4x)/3)^3.) once you have the derivative, plug in the x value. This will generate the slope of the tangent line at that point. Now with a slope, and a point on the line you can use point slope form to write an equation. the equation of your derivative : y' =1-(5x^4) then plug in x : y' = 1-5(1^4) =1-5 = -4 therefore -4 equals the slope of the tangent line at your point Then use point slope form: y-y1 = m(x-x1) y-0 = -4(x-1) y= -4x +4