Question

Find the slope of the tangent line to the graph of the given function at the given value of x. Find the equation of the tangent line.

y=x^4-2x^3+3; x=2

How would the slope of a tangent line be determined with the given​ information?

A. Set the derivative equal to zero and solve for x.

B. Substitute values of x into the equation and solve for y. Plot the resulting points to find the linear equation.

C.Substitute 2 for x into the derivative of the function and evaluate.

D.Substitute values of y into the equation and solve for x. Plot the resulting points to find the linear equation.

**The slope of the tangent line is

**The equation of the line is

​(Type an equation. Type your answer in​ slope-intercept form.)

Answer 1

Answer: The correct option is (C). The slope of the tangent line is 8 and equation of tangent line is y=8x-13

Explanation: the slope, m, of the line tangent to the graph of a function, f(x) is defined as

----------------------------------- (1)

The given function is y=x⁴-2x³+3

So, the slope, m will be

---------------------------------------- (2)

at x=2,

m=4*(2³)-6*(2²)=4*8-6*4=32-24=8

So, slope, m=8

Now, at x=2, y=2⁴-2*2³+3=16-16+3=3

So, the points are (x1, y1)=(2, 3)

Now, the equation of line with slope, m=8 and passing through points (x1, y1)=(2,3) will be given by

y-y1=m(x-x1) --------------------------------------------------- (3)

substitute x1=2 and y1=3 into eqn 3

y-3=8(x-2)

y-3=8x-16

y=8x-16+3

y=8x-13

This is the required equation of tangent line with slope-intercept form. Slope=8 and intercept=-13.