Precalculus - HW #29
The Slope & Equation of the Tangent
March 28-29, 2017
I.
For the function f(x)
Name:_______________ Per:_____
= 2x2 - 20x + 43
(a) Find the difference quotient for f(x).
(b) Confirm that the derivative for f(x) is f’(x) = 4x - 20.
(c) Find the slope of the line between the following pairs of points two different ways: using the
slope formula and using the difference quotient.
(i) (9,f(9)) & (10,f(10))
(ii) (9,f(9)) & (9.1,f(9.1))
(iii) (9,f(9)) & (9.01,f(9.01))
Use the derivative f’(x) to find the “instantaneous slope” of f(x) at the point (9,f(9)).
(d)
(e) Find the equation of the line tangent to f(x) at the point (9,f(9)). Hint: this is the line passing
through (9,f(9)) with a slope of f’(9).
(f) Use a graphing utility to graph f(x) and the tangent line you found in part (e). Sketch the graph
making sure to label the POI, which should be (9,f(9)). Include window dimensions.
(g) Find the equation of the line tangent to f(x) at the point (2,f(2)). Hint: this is the line passing
through (2,f(2)) with a slope of f’(2).
(h) Use a graphing utility to graph f(x) and the tangent line you found in part (g). Sketch the graph
making sure to label the POI, which should be (2,f(2)). Include window dimensions.
II.
For the function g(x)
= x3 - 12x + 21
(a) Confirm that the derivative for g(x) is g’(x) = 3x2 - 12
(b) Find the line tangent to g(x) at the following points.
(i) (5,g(5))
(ii) (2,g(2))
(iii) (-1,g(-1))
(c) Use a graphing utility to graph g(x) and the tangent lines you found in part (b) (separate graphs).
Make a sketch for each graph. Make sure to label the POI and include window dimensions.
(d) The point (2,g(2)) is a relative minimum for g(x). What is the slope of the line tangent to this
point (i.e. the value of the derivative at the point: g’(2))? Use the derivative to symbolically
find the x-coordinate for the relative maximum of g(x). Then find the y-coordinate for the
relative maximum.
III.
For the function h(x)
(a)
(b)
(c)
(d)
= -3x2 - 42x - 120
Confirm that the derivative for g(x) is h’(x) = -6x
- 42
Use the derivative to find the x-coordinate for the vertex. Then find the y-coordinate.
Change h(x) into standard form by completing the square.
Graph h(x) using the 5-point graphing method. You’ve already found the vertex. Next
find the x-intercepts, y-intercept and reflection point. Show complete work for each step.