# Find the function f given that the slope of the tangent line at any point (x, f(x)) is f '(x) and that the graph of f passes through the given point. f '(x)=5(2x − 1)^{4}, (1, 9)

Question
Functions
Find the function f given that the slope of the tangent line at any point (x, f(x)) is f '(x) and that the graph of f passes through the given point. $$f '(x)=5(2x − 1)^{4}, (1, 9)$$

2021-03-06

Start by finding the slope of f(x). To do this, integrate f'(x). The integral comes out to $$\frac{(2x-1)^{5}}{2}+C$$. Now solve for C by solving $$f(1)=\frac{(2\times1−1)^{5}}{2}+C=9$$, which gives us a C value of $$\frac{17}{2}$$, thus the function $$f(x)=\frac{(2x−1)^{5}}{2}+\frac{17}{2}$$

### Relevant Questions

Consider the curves in the first quadrant that have equationsy=Aexp(7x), where A is a positive constant. Different valuesof A give different curves. The curves form a family,F. Let P=(6,6). Let C be the number of the family Fthat goes through P.
A. Let y=f(x) be the equation of C. Find f(x).
B. Find the slope at P of the tangent to C.
C. A curve D is a perpendicular to C at P. What is the slope of thetangent to D at the point P?
D. Give a formula g(y) for the slope at (x,y) of the member of Fthat goes through (x,y). The formula should not involve A orx.
E. A curve which at each of its points is perpendicular to themember of the family F that goes through that point is called anorthogonal trajectory of F. Each orthogonal trajectory to Fsatisfies the differential equation dy/dx = -1/g(y), where g(y) isthe answer to part D.
Find a function of h(y) such that x=h(y) is the equation of theorthogonal trajectory to F that passes through the point P.
For the equation (-1,2), $$y= \frac{1}{2}x - 3$$, write an equation in slope intercept form for the line that passes through the given point and is parallel to the graph of the given equation.
Find the absolute maximum and absolute minimum values of f on the given interval.
$$f(x)=5+54x-2x^{3}, [0,4]$$
Consider the function $$f(x)=2x^{3}+6x^{2}-90x+8, [-5,4]$$
find the absolute minimum value of this function.
find the absolute maximum value of this function.
Consider the function $$f(x)=2x^{3}-6x^{2}-18x+9$$ on the interval [-2,4].
What is the absolute minimum of f(x) on [-2,4]?
What is the absolute maximum of f(x) on [-2,4]?
Find the equation of the tangent plane to the graph of $$\displaystyle{f{{\left({x},{y}\right)}}}={8}{x}^{{{2}}}-{2}{x}{y}^{{{2}}}$$ at the point (5,4).
A)z=23x-15y+42
B)z=48x-80y+120
C)0=48x-80y+120
D)0=23x-15y+42
Find the absolute maximum value and the absolute minimum value, if any, of the function.
$$f(x)=8x-\frac{9}{x}$$
Use the given conditions to write an equation for the line in point-slope form and in slope-intercept form.
Passing through (3,-1) and perpendicular to the line whose equation is x-9y-5=0
Write an equation for the line in point-slope form and slope-intercept form.
The graph of y = f(x) contains the point (0,2), $$\displaystyle{\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}}={\frac{{-{x}}}{{{y}{e}^{{{x}^{{2}}}}}}}$$, and f(x) is greater than 0 for all x, then f(x)=
A) $$\displaystyle{3}+{e}^{{-{x}^{{2}}}}$$
B) $$\displaystyle\sqrt{{{3}}}+{e}^{{-{x}}}$$
C) $$\displaystyle{1}+{e}^{{-{x}}}$$
D) $$\displaystyle\sqrt{{{3}+{e}^{{-{x}^{{2}}}}}}$$
E) $$\displaystyle\sqrt{{{3}+{e}^{{{x}^{{2}}}}}}$$
$$g(x)=-x^{2}+2x+6$$