WebThe maximum and minimum values for sin(x) is 1 and -1. The value of sin^2(x) at these points is 1. Sticking the maximum value of sin(x) in the equation you get the maximum of 1 + 4*1 -1 = 4. WebThe table below shows various graphs of f(x) and tangent lines at points x 1, x 2, and x 3. Since f'(x) is the slope of the line tangent to f(x) at point x, the concavity of f(x) can be …

Definition of Concavity & How to Test for It Tangent Line …

WebSimilarly, the righthand plot in Figure1.87 depicts a function that is concave down; in this case, we see that the tangent lines alway lie above the curve and that the slopes of the tangent lines are decreasing as we move from left to right. The fact that its derivative, \(f'\text{,}\) is decreasing makes \(f\) concave down on the interval shown. WebTranscribed Image Text: (a) Find a function f that has y = 4 – 3x as a tangent line and whose derivative is equal to ƒ' (x) = x² + 4x + 1. (b) Find the area under the curve for f (x) = x³ on [−1, 1]. e2t - 2 (c) Determine where the function is f (x) = cos (t²-1) + 3 (d) Express ² sin (x²) dx as limits of Riemann sums, using the right ... ditched cargo crossword

Justification using second derivative (article) Khan Academy

WebNov 2, 2024 · Example \(\PageIndex{2}\): Finding a Tangent Line. Find the equation of the tangent line to the curve defined by the equations \[x(t)=t^2−3, \quad y(t)=2t−1, \quad\text{for }−3≤t≤4 \nonumber \] when \(t=2\). Solution. First find the slope of the tangent line using Equation \ref{paraD}, which means calculating \(x′(t)\) and \(y′(t)\): WebA tangent line to a curve lies above the curve if it is concave down, and it lies below the curve if it is concave up. Here, let us examine a function f(x) that is concave down … WebThe plots in Figure2.108 highlight yet another important thing that we can learn from the concavity of the graph near the point of tangency: whether the tangent line lies above or below the curve itself. This is key because … ditched conference