For each of the following curves and given point:
Find the equation of the tangent.
f \left( x \right) = 2 e^{x} at the point where x = 0.
f \left( x \right) = 5e^{4x} at the point (0, 5).
f \left( x \right) = \dfrac{e^{x}}{6} at the point \left(0, \dfrac{1}{6}\right).
f \left( x \right) = 4 e^{3x} - 2x at the point where x=0.
For each of the following curves, at the given point:
Find the exact gradient of the normal.
y=\dfrac{e^x}{x} at the point where x=-1.
y=xe^x at the point where x=1.
y=x-e^{-x} at the point where x=2.
The tangent to the curve f \left( x \right) = 2.5 e^{x} at the point x = 7.5, is parallel to the tangent to the curve g \left( x \right) = e^{ 2.5 x} at the point where x = k.
Find the value of k.
Consider y = x e^{ - x }:
Find the gradient function.
Find the gradient of the normal at the point \left(3, 3 e^{ - 3 }\right).
Hence, find the equation of the normal at the point \left(3, 3 e^{ - 3 }\right).
The tangent to the curve y=x^{2}e^{x} at x=1, cuts the axes at points A and B .
Find the exact gradient of the tangent at x=1.
Consider the functions y = 3 e^{-x} and y = 2 + e^{-x}:
Find the point of intersection of the functions.
Find the gradient of the tangent to y = 3 e^{-x} at the point of intersection.
Determine the acute angle this tangent makes with the x-axis, to the nearest degree.
Find the gradient of the tangent to y = 2 + e^{-x} at the point of intersection.
Determine the acute angle this tangent makes with the x-axis, to the nearest degree.
Find the acute angle between the two tangents.
Consider the functions g \left( x \right) = e^{ 7 x} and f \left( x \right) = e^{ - 7 x }.
Consider the tangent line to g \left( x \right) at x = 0. Find the acute angle that the tangent line makes with the x-axis, correct to one decimal place.
Consider the tangent line to f \left( x \right) at x = 0. Find the acute angle that the tangent line makes with the x-axis, correct to one decimal place.
Hence, determine the acute angle between the two tangents at x = 0.
Consider the curve f \left( x \right) = e^{x} + e x. Show that the tangent to the curve at the point \left(1, 2 e\right) passes through the origin.
Consider the function f \left( x \right) = 3 - e^{ - x }.
Find f \left( 0 \right).
Find f' \left( 0 \right).
State whether f (x) is an increasing or decreasing function. Explain your answer.
Find the limit of f (x) as x \to \infty.
Consider the function f \left( t \right) = \dfrac{4}{2 + 3 e^{ - t }}.
Find f \left( t \right) when t = 0.
Find f' \left( t \right).
State whether f(x) is an increasing or decreasing function. Explain your answer.
Hence, state how many stationary points the function has.
Find the limit of f (t) as t \to \infty.
Consider the function y = e^{x} \left(x - 3\right).
Find the coordinates of the turning point.
State whether this is a minimum or maximum turning point.
Consider the function f \left( x \right) = 4 e^{ - x^{2} }.
Find f' \left( x \right).
Find the values of x for which:
f' \left( x \right) \gt 0.
f' \left( x \right) \lt 0
Find the limit of f(x) as x \to \infty.
Find the limit of f(x) as x \to - \infty.
Sketch the graph of f \left( x \right).
Consider the function f \left( x \right) = e^{x} - 2 e^{ - x }.
Find the x-intercept, to two decimal places.
Find the y-intercept.
What value does y approach as x \to \infty.
What value does y approach as x \to - \infty.
Sketch the graph of the function.
For each of the following functions:
Find the x-intercept.
Find the y-intercept.
State the nature of the stationary points.
Find any points of inflection.
Describe what happens to the function as x \to \infty.
Sketch the graph of the function.
y = e^{2x} - 3 e^{x}
Consider the function y = x^{3} e^{-x}:
Find the x-intercept.
Find the y-intercept.
Find y \rq.
Find the coordinates of any stationary points, correct to two decimal places if necessary.
Describe the nature of the stationary points.
Sketch the graph of the function.
For the function y = e^{-x}\sqrt{x}:
Find the x-intercept.
Find the y-intercept.
Find y \rq.
Find the coordinates of any stationary points, correct to two decimal places.
State the nature of the stationary points.
Sketch the graph of the function.
Consider the function y = e^{ 2 x} - 3 e^{x}.
Find the x-intercept.
Find the y-intercept.
Find the coordinates of any stationary points, correct to two decimal places.
State the nature of the stationary point.
Describe the behaviour of this function as x \to \infty and x \to -\infty.
Sketch the graph of the function.