Questions: Use the function below to answer parts (a)-(d).
f(x) = (5/2) x^2
(a) Graph the function
(b) Draw lines tangent to the graph at points whose x-coordinates are 0, and 1,3.
(c) Find f'(x) by determining lim (f(x+h)-f(x))/h.
(d) Find f'(0), f'(1), and f'(3).
Transcript text: Use the function below to answer parts (a)-(d).
$f(x)=\frac{5}{2} x^{2}$
(a) Graph the function
(b) Draw lines tangent to the graph at points whose $x$-coordinates are 0 , and 1,3 .
(c) Find $f^{\prime}(x)$ by determining $\lim \frac{f(x+h)-f(x)}{h}$.
(d) Find $f^{\prime}(0), f^{\prime}(1)$, and $f^{\prime}(3)$.
Solution
Solution Steps
Step 1: Graphing the function
The function f(x)=25x2 is a parabola that opens upwards and is vertically stretched by a factor of 5/2. The vertex is at the origin (0,0). Graph C correctly represents this function.
Step 2: Drawing tangent lines and determining f'(x)
The formula for the derivative provided is the limit definition of a derivative. For f(x)=25x2, the derivative is f′(x)=5x. Graph B correctly depicts the tangent lines at x=0, x=1, and x=3. At x = 0, the tangent line is horizontal (slope = 0). At x=1, the tangent line has a positive slope, and at x=3, the tangent line has a steeper positive slope.
Step 3: Finding f'(0), f'(1), and f'(3)
Using f′(x)=5x, we find the following values:
f′(0)=5(0)=0f′(1)=5(1)=5f′(3)=5(3)=15