If f(2)=1 and f(2+h)=(h+1)3, compute f′(2).
If f(−1)=5 and f(−0.9)=5.2, estimate f′(−1).
If the line y=−3x+2 is tangent to f(x) at x=−4, find f(−4). Your answer should be expressed as an integer.
If the line y=x−1 is tangent to the graph of f(x) at x=−3, find f′(−3).
Use the definition of a derivative:
f'(x) = lim_{∆x→0} (f(x+∆x)-f(x))/∆x
Like so:
f(2) = 1
f(2+h) = (h+1)^3
f'(2) = lim_{h→0} (f(2+h)-f(2))/h
f'(2) = lim_{h→0} ((h+1)^3 -1)/h
f'(2) = lim_{h→0} (h^3 + 3h^2 + 3h)/h
f'(2) = lim_{h→0} (h^2 + 3h + 3)
f'(2) = 3
Thanks, can you tell me the rest?
To find the value of the derivative of a function at a given point, we can use the definition of the derivative. The derivative of a function f(x) at a point x=a is defined as the limit of the difference quotient as h approaches 0:
f'(a) = lim(h→0) [f(a+h) - f(a)] / h
Let's use this definition to find the derivative in each of the given cases.
1. If f(2) = 1 and f(2+h) = (h+1)^3, we can find f'(2) by evaluating the limit as h approaches 0:
f'(2) = lim(h→0) [(2+h+1)^3 - 1] / h
= lim(h→0) [(h+3)^3 - 1] / h
= lim(h→0) [(h^3 + 9h^2 + 27h + 27 - 1] / h
= lim(h→0) [(h^3 + 9h^2 + 27h + 26] / h
Now, we can simplify the expression and evaluate the limit:
f'(2) = lim(h→0) [h^2 + 9h + 27 + 26/h]
= lim(h→0) [h^2 + 9h + 27] + lim(h→0) [26/h]
= 0^2 + 9(0) + 27 + ∞
= 27 + ∞
= ∞
Therefore, the value of f'(2) is infinity.
2. If f(-1) = 5 and f(-0.9) = 5.2, we can estimate f'(-1) using the definition of the derivative again:
f'(-1) ≈ [(f(-1+h) - f(-1)) / h]
We need to choose a small value of h to estimate the derivative. Let's take h = 0.1:
f'(-1) ≈ [f(-0.9) - f(-1)] / 0.1
≈ [5.2 - 5] / 0.1
≈ 0.2 / 0.1
≈ 2
Therefore, the estimated value of f'(-1) is 2.
3. If the line y = -3x + 2 is tangent to f(x) at x = -4, we know that the slope of the tangent line is equal to the derivative of f(x) at x = -4.
So, f'(-4) = -3.
4. If the line y = x - 1 is tangent to the graph of f(x) at x = -3, we can find f'(-3) by comparing the slope of the tangent line (1) to the derivative of f(x) at x = -3.
So, f'(-3) = 1.
I hope this explanation helps you understand how to approach these types of problems using the definition of the derivative or the information given in the question.