Given cosh x = 1/2 (e^x + e^-x) and sinh x = 1/2 (e^x - e^-x), Use these definitions to show cosh2 x - sinh2…

Question

Given cosh x = 1/2 (e^x + e^-x) and sinh x = 1/2 (e^x - e^-x), Use these definitions to show cosh2 x - sinh2 x = 1

ScanSolve AI Tutor · Accepted Answer

Here's how to solve question 5(b): Step 1: Define the function and its derivative. Let y = f(x) = x^(1/3). The derivative is: dy/dx = f'(x) = 1/3x^(1/3−1) = 1/3x^(−2/3) = 13x^(2/3) Step 2: Choose a suitable value for x and calculate Δ x. To approximate √[3]1003, we choose a value of x close to 1003 for which the cube root is easy to calculate. Let x = 1000. Then the change in x is Δ x = 1003 − 1000 = 3. Step 3: Evaluate f(x) and f'(x) at x = 1000. f(1000) = 1000^(1/3) = (10³)^(1/3) = 10; f'(1000) = 13(1000)^(2/3) = 13((10³)^(1/3))² = (1)/(3(10)²) = (1)/(3 × 100) = 1/300 Step 4: Use the approximation formula. The approximate value of f(x + Δ x) is given by f(x) + f'(x) Δ x. √[3]1003 ≈ f(1000) + f'(1000) × Δ x √[3]1003 ≈ 10 + (1/300) × 3 √[3]1003 ≈ 10 + 3/300 √[3]1003 ≈ 10 + 1/100 √[3]1003 ≈ 10 + 0.01 √[3]1003 ≈ 10.01

Given cosh x = 1/2 (e^x + e^-x) and sinh x = 1/2 (e^x - e^-x), Use these definitions to show cosh2 x - sinh2 x = 1

Answer

Step-by-step solution

Define the function and its derivative.

Choose a suitable value for $x$ and calculate $\Delta x$ .

Evaluate $f(x)$ and $f'(x)$ at $x = 1000$ .

Use the approximation formula.

Try one yourself

Still stuck on this one?

More Mathematics Questions

Given cosh x = 1/2 (e^x + e^-x) and sinh x = 1/2 (e^x - e^-x), Use these definitions to show cosh2 x - sinh2 x = 1

Answer

Step-by-step solution

Define the function and its derivative.

Choose a suitable value for xxx and calculate Δx\Delta xΔx.

Evaluate f(x)f(x)f(x) and f′(x)f'(x)f′(x) at x=1000x = 1000x=1000.

Use the approximation formula.

Try one yourself

Still stuck on this one?

More Mathematics Questions

Choose a suitable value for $x$ and calculate $\Delta x$ .

Evaluate $f(x)$ and $f'(x)$ at $x = 1000$ .