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Math  /  Calculus

Questionإذا كانت lim⁡x→1(f(x)−f(1)x−1)=10 ركت f(x)=mx2\lim _{x \rightarrow 1}\left(\frac{f(x)-f(1)}{x-1}\right)=10 \text { ركت } f(x)=m x^{2}
فإن قيمة m تساوي اخترأحد الخيارات a. 2 b. 5 c. 3 d. 1

Studdy Solution

STEP 1

1. We are given the limit: lim⁡x→1(f(x)−f(1)x−1)=10\lim _{x \rightarrow 1}\left(\frac{f(x)-f(1)}{x-1}\right)=10.
2. The function is given as f(x)=mx2 f(x) = mx^2 .
3. We need to find the value of m m .

STEP 2

1. Substitute f(x)=mx2 f(x) = mx^2 into the limit expression.
2. Calculate f(1) f(1) .
3. Simplify the limit expression.
4. Solve for m m using the given limit value.

STEP 3

Substitute f(x)=mx2 f(x) = mx^2 into the limit expression:
lim⁡x→1(mx2−f(1)x−1)=10 \lim _{x \rightarrow 1}\left(\frac{mx^2 - f(1)}{x-1}\right) = 10

STEP 4

Calculate f(1) f(1) :
f(1)=m(1)2=m f(1) = m(1)^2 = m

STEP 5

Substitute f(1)=m f(1) = m into the limit expression:
lim⁡x→1(mx2−mx−1)=10 \lim _{x \rightarrow 1}\left(\frac{mx^2 - m}{x-1}\right) = 10
Factor out m m from the numerator:
lim⁡x→1(m(x2−1)x−1)=10 \lim _{x \rightarrow 1}\left(\frac{m(x^2 - 1)}{x-1}\right) = 10
Recognize that x2−1 x^2 - 1 can be factored as (x−1)(x+1) (x-1)(x+1) :
lim⁡x→1(m(x−1)(x+1)x−1)=10 \lim _{x \rightarrow 1}\left(\frac{m(x-1)(x+1)}{x-1}\right) = 10
Cancel the (x−1) (x-1) terms:
lim⁡x→1(m(x+1))=10 \lim _{x \rightarrow 1}\left(m(x+1)\right) = 10

STEP 6

Evaluate the limit as x→1 x \rightarrow 1 :
m(1+1)=10 m(1+1) = 10 2m=10 2m = 10
Solve for m m :
m=102 m = \frac{10}{2} m=5 m = 5
The value of m m is:
5 \boxed{5}

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